A function f(x) is defined by:
\(f(x)=\begin{cases} x+2, & \text{if } x \lt 0\\ -x+2, & \text{if } x \gt 0 \end{cases}\)
Which of the following is true?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → f(x) is a continuous function on its domain.
The function is defined for x < 0 and for x > 0 only, so its domain is all real numbers except 0.
For x < 0, f(x) = x + 2 is a polynomial, so it is continuous at every such point.
For x > 0, f(x) = −x + 2 is a polynomial, so it is continuous at every such point.
A function is called continuous when it is continuous at every point of its domain. So f is a continuous function on its domain.
Why the other options are incorrect
Option (1) f(x) is discontinuous at x = 0. — Incorrect. Continuity is checked only at points of the domain, and 0 is not in the domain, so the question does not arise at x = 0.
Option (2) f(x) is not a function as f(0) is not defined. — Incorrect. A function need not be defined at every real number; here it is a function with domain all real x except 0.
Option (4) f(x) is a discontinuous function on its domain. — Incorrect. f is continuous at every point where it is defined.
NCERT (2026-27 edition):
Continuity
