Check whether function f(x) defined as f(x) = cases ( x-3 )/(2(x-3)), & x < 3 \\ (x-6)/(6), & x ≥ 3 cases is continuous at x = 3 or not ?

MathsCBSE Class 12 · Board exam 2026, set 65/1/1 · 2 marks · Very short answer

CBSE Class 12 Mathematics 2026: question 21(a)

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, section B, question 21(a), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q21(a) · 2 marks · Very short answerContinuity and Differentiability

Check whether function $f (x)$ defined as
$f (x) = {\frac{∣ x - 3∣}{2 ( x - 3 )}, \frac{x - 6}{6}, x < 3 x \geq 3$
is continuous at $x = 3$ or not ?

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/1/1.

Hints

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

Hint 1

What sign does $x - 3$ have when $x < 3$ ? Use it to remove the modulus.

Hint 2

Find $f (3)$ , the left-hand limit and the right-hand limit separately.

Hint 3

A function is continuous at a point when all three values are equal.

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.

For $x < 3$ , $x - 3 < 0$ , so $∣ x - 3∣ = - (x - 3)$ . Then $f (x) = \frac{- ( x - 3 )}{2 ( x - 3 )} = - \frac{1}{2}$ for $x < 3$ , and $f (x) = \frac{x - 6}{6}$ for $x \geq 3$ .
1 mark
$f (3) = \frac{3 - 6}{6} = - \frac{1}{2}$ .
LHL $= lim_{x \to 3^{-}} (- \frac{1}{2}) = - \frac{1}{2}$ .
RHL $= lim_{x \to 3^{+}} \frac{x - 6}{6} = \frac{- 3}{6} = - \frac{1}{2}$ .
½ mark
Since LHL = RHL = $f (3) = - \frac{1}{2}$ , $f$ is continuous at $x = 3$ .
½ mark

Answer: f is continuous at x = 3 (LHL = RHL = f(3) = −1/2)

Where the marks are

Removes the modulus for $x < 3$ to get $f (x) = - \frac{1}{2}$ (1 mark)
Finds $f (3)$ and both one-sided limits equal to $- \frac{1}{2}$ (½ mark)
Concludes $f$ is continuous at $x = 3$ (½ 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 paper2026 board paper, set 65/1/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2026: question 21(a)

Hints

Step-by-step solution

Revise the chapter

More questions from Continuity and Differentiability