If f(x)= x + x-1 , then which of the following is correct ? (A) f(x) is both continuous and differentiable, at x=0 and x=1. (B) f(x) is differentiable but not continuous, at x=0 and x=1. (C) f(x) is continuous but not differentiable, at x=0 and x=1. (D) f(x) is neither continuous nor differentiable, at x=0 and x=1.

MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 1 mark · MCQ

CBSE Class 12 Mathematics 2025: question 5

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section A, question 5, for 1 mark. Try it first; the hints and our step-by-step solution are below.

Q5 · 1 mark · MCQContinuity and Differentiability

If $f (x) = ∣ x ∣ + ∣ x - 1∣$ , then which of the following is correct ?

(A) $f (x)$ is both continuous and differentiable, at $x = 0$ and $x = 1$ .
(B) $f (x)$ is differentiable but not continuous, at $x = 0$ and $x = 1$ .
(C) $f (x)$ is continuous but not differentiable, at $x = 0$ and $x = 1$ .
(D) $f (x)$ is neither continuous nor differentiable, at $x = 0$ and $x = 1$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Write $f$ piecewise on $x < 0$ , $0 \leq x \leq 1$ and $x > 1$ .

Hint 2

Compare the slopes on either side of 0 and of 1.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

The modulus functions are continuous everywhere, so $f$ is continuous at both points.
At $x = 0$ : LHD $= - 2$ , RHD $= 0$ . At $x = 1$ : LHD $= 0$ , RHD $= 2$ . The one-sided derivatives differ, so $f$ is not differentiable at either point. Hence option (C).
1 mark

Answer: (C)

Where the marks are

Notes continuity and shows the left and right derivatives differ at 0 and 1 (1 mark)

Revise the chapter

Chapter: Continuity and Differentiability (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Continuity and Differentiability, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 5

Hints

Step-by-step solution

Revise the chapter

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