Class 12 Maths · Original EZT practice
Continuity and Differentiability (Class 12 Maths Chapter 5): when a curve has no breaks and no corners
Differentiation runs every optimisation and rate problem ahead of you - but it only works where a function is smooth: no breaks (continuous), no sharp corners (differentiable). This chapter gives you the tests and the four derivative rules. Below: the ideas in plain words, then original EZT practice with worked answers. (Note: this is original practice, not a textbook exercise key.)
11 original worked examples, 11 fresh self-check questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 17 October 2026:
1. Continuity: no jumps, no holes
A function is continuous at a point if the limit as x approaches it equals the function's value there: the graph passes through with no break. Polynomials, sine, cosine and e^x are continuous everywhere.
2. Differentiability implies continuity (not the reverse)
If a function has a derivative at a point, it must be continuous there - but continuity alone is not enough: a sharp corner blocks the derivative.
3. The four derivative rules
Sum rule: differentiate term by term. Product rule: (uv)' = u'v + uv'. Quotient rule: (u/v)' = (u'v - uv')/v2. Chain rule: differentiate the outside, multiply by the derivative of the inside.
4. Rolle's theorem and the mean value theorem
Rolle's theorem: if a continuous, differentiable function has equal values at two ends of an interval, somewhere between them the tangent is flat (f'(c) = 0). The mean value theorem generalises it: somewhere the tangent slope equals the average slope across the interval.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Is f(x) = x2 continuous at x = 2?
Choose the method: A function is continuous at a point if the limit as x approaches it equals the function's value there: the graph passes through with no break.
EZT practice question
Is f(x) = x2 continuous at x = 2?
Show the worked solution
- Identify the function, point or interval and the theorem or derivative rule that applies.
- Compute the limit, derivative or average slope carefully.
- Verify the required condition before stating the conclusion.
Answer: the limit as x approaches 2 is 4, and f(2) = 4 - equal, so continuous.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not assume continuity alone gives differentiability; check corners, holes and the theorem conditions first.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
f(x) = (x2 - 1)/(x - 1) is undefined at x = 1. Is it continuous there?
Choose the method: A function is continuous at a point if the limit as x approaches it equals the function's value there: the graph passes through with no break.
EZT practice question
f(x) = (x2 - 1)/(x - 1) is undefined at x = 1. Is it continuous there?
Show the worked solution
- Identify the function, point or interval and the theorem or derivative rule that applies.
- Compute the limit, derivative or average slope carefully.
- Verify the required condition before stating the conclusion.
Answer: no - it has no value at x = 1 (a hole), even though the limit exists (it equals 2); continuity needs limit AND value to match.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not assume continuity alone gives differentiability; check corners, holes and the theorem conditions first.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
f(x) = |x| at x = 0: continuous?
Choose the method: If a function has a derivative at a point, it must be continuous there - but continuity alone is not enough: a sharp corner blocks the derivative.
EZT practice question
f(x) = |x| at x = 0: continuous? differentiable?
Show the worked solution
- Identify the function, point or interval and the theorem or derivative rule that applies.
- Compute the limit, derivative or average slope carefully.
- Verify the required condition before stating the conclusion.
Answer: continuous (the V has no break) but not differentiable - the left slope (-1) and right slope (+1) disagree at the corner.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not assume continuity alone gives differentiability; check corners, holes and the theorem conditions first.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Differentiate x3
Choose the method: Sum rule: differentiate term by term.
EZT practice question
Differentiate x3.
Show the worked solution
- Identify the function, point or interval and the theorem or derivative rule that applies.
- Compute the limit, derivative or average slope carefully.
- Verify the required condition before stating the conclusion.
Answer: 3x2 (power rule).
Check: The substitution and final condition match the worked statement.
Watch for this: Do not assume continuity alone gives differentiability; check corners, holes and the theorem conditions first.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Differentiate x2 sin x
Choose the method: Sum rule: differentiate term by term.
EZT practice question
Differentiate x2 sin x.
Show the worked solution
- Identify the function, point or interval and the theorem or derivative rule that applies.
- Compute the limit, derivative or average slope carefully.
- Verify the required condition before stating the conclusion.
Answer: product rule: 2x sin x + x2 cos x.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not assume continuity alone gives differentiability; check corners, holes and the theorem conditions first.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Differentiate x/(x + 1)
Choose the method: Sum rule: differentiate term by term.
EZT practice question
Differentiate x/(x + 1).
Show the worked solution
- Identify the function, point or interval and the theorem or derivative rule that applies.
- Compute the limit, derivative or average slope carefully.
- Verify the required condition before stating the conclusion.
Answer: quotient rule: [(1)(x + 1) - x(1)]/(x + 1)2 = 1/(x + 1)2.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not assume continuity alone gives differentiability; check corners, holes and the theorem conditions first.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Differentiate sin(3x + 2)
Choose the method: Sum rule: differentiate term by term.
EZT practice question
Differentiate sin(3x + 2).
Show the worked solution
- Identify the function, point or interval and the theorem or derivative rule that applies.
- Compute the limit, derivative or average slope carefully.
- Verify the required condition before stating the conclusion.
Answer: chain rule: cos(3x + 2) x 3 = 3 cos(3x + 2).
Check: The substitution and final condition match the worked statement.
Watch for this: Do not assume continuity alone gives differentiability; check corners, holes and the theorem conditions first.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Differentiate x2 e^x
Choose the method: Sum rule: differentiate term by term.
EZT practice question
Differentiate x2 e^x.
Show the worked solution
- Identify the function, point or interval and the theorem or derivative rule that applies.
- Compute the limit, derivative or average slope carefully.
- Verify the required condition before stating the conclusion.
Answer: product rule: 2x e^x + x2 e^x = e^x(2x + x2).
Check: The substitution and final condition match the worked statement.
Watch for this: Do not assume continuity alone gives differentiability; check corners, holes and the theorem conditions first.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
What is the derivative of ln x?
Choose the method: Sum rule: differentiate term by term.
EZT practice question
What is the derivative of ln x?
Show the worked solution
- Identify the function, point or interval and the theorem or derivative rule that applies.
- Compute the limit, derivative or average slope carefully.
- Verify the required condition before stating the conclusion.
Answer: 1/x (and the derivative of e^x is e^x itself).
Check: The substitution and final condition match the worked statement.
Watch for this: Do not assume continuity alone gives differentiability; check corners, holes and the theorem conditions first.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Verify Rolle's theorem for f(x) = x2 - 4 on [-2, 2]
Choose the method: Rolle's theorem: if a continuous, differentiable function has equal values at two ends of an interval, somewhere between them the tangent is flat (f'(c) = 0).
EZT practice question
Verify Rolle's theorem for f(x) = x2 - 4 on [-2, 2].
Show the worked solution
- Identify the function, point or interval and the theorem or derivative rule that applies.
- Compute the limit, derivative or average slope carefully.
- Verify the required condition before stating the conclusion.
Answer: f(-2) = f(2) = 0, so some c has f'(c) = 0; f'(x) = 2x gives c = 0, which lies in (-2, 2) - verified.
Check: The substitution and final condition match the worked statement.
Watch for this: Do not assume continuity alone gives differentiability; check corners, holes and the theorem conditions first.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Apply the mean value theorem to f(x) = x2 on [1, 3]
Choose the method: Rolle's theorem: if a continuous, differentiable function has equal values at two ends of an interval, somewhere between them the tangent is flat (f'(c) = 0).
EZT practice question
Apply the mean value theorem to f(x) = x2 on [1, 3].
Show the worked solution
- Identify the function, point or interval and the theorem or derivative rule that applies.
- Compute the limit, derivative or average slope carefully.
- Verify the required condition before stating the conclusion.
Answer: average slope = (9 - 1)/(3 - 1) = 4; f'(c) = 2c = 4 gives c = 2, inside (1, 3) - verified. TRY IT (no answers, for practice): (a) Is f(x) = 1/x continuous at x = 0? (b) Differentiate x4 + 3x. (c) Differentiate (sin x)/x. (d) Does Rolle's theorem apply to |x| on [-1, 1]? Say why. Edition note: chapter numbering (5) matches the pre-rationalisation book for this chapter. FAQ (FAQPage schema, 5 Q&A): Q1: When is a function continuous at a point? A: when the limit as x approaches the point exists and equals the function's value there - no jump, no hole. Q2: Does continuity imply differentiability? A: no - differentiability implies continuity, but a continuous function with a sharp corner, like |x| at 0, has no derivative there. Q3: What is the chain rule? A: for a function inside a function, differentiate the outer one first and multiply by the derivative of the inner one - sin(3x + 2) becomes 3 cos(3x + 2). Q4: What does Rolle's theorem say? A: a continuous, differentiable function with equal end-values has at least one interior point where the derivative is zero - a flat tangent. Q5: Are these the NCERT exercise questions? A: No. EZT pages are original practice on this chapter - use them to prepare for the textbook exercises, not as a substitute for your book. Internal links: /class-12-maths-chapter-1-relations-and-functions (live), /class-11-maths-chapter-1-sets (live), /tuitions, /home-tuition-fees-delhi-ncr Schema: LearningResource + FAQPage, same component pattern as existing solutions pages Mobile: responsive
Check: The substitution and final condition match the worked statement.
Watch for this: Do not assume continuity alone gives differentiability; check corners, holes and the theorem conditions first.
Now try it yourself
Create one new example with different numbers and solve it by the same method.
Check your answer
Keep the final result in your notebook; the page-level Try It prompts are intentionally answer-free.
The worked answer above shows the method; independent practice should not become another answer key.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Is f(x) = 1/x continuous at x = 0?
- Differentiate x4 + 3x.
- Differentiate (sin x)/x.
- Does Rolle's theorem apply to |x| on [-1, 1]? Say why.
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Questions about this practice page
- When is a function continuous at a point?
- when the limit as x approaches the point exists and equals the function's value there - no jump, no hole.
- Does continuity imply differentiability?
- no - differentiability implies continuity, but a continuous function with a sharp corner, like |x| at 0, has no derivative there.
- What is the chain rule?
- for a function inside a function, differentiate the outer one first and multiply by the derivative of the inner one - sin(3x + 2) becomes 3 cos(3x + 2).
- What does Rolle's theorem say?
- a continuous, differentiable function with equal end-values has at least one interior point where the derivative is zero - a flat tangent.
- Are these the NCERT exercise questions for Continuity and Differentiability?
- No. EZT pages are original practice on this chapter - use them to prepare for the textbook exercises, not as a substitute for your book.