Question

Let $[x]$ denotes the integral part of $x \in R$ and $g(x) = x - [x]$. Let $f(x)$ be any continuous function with $f(0) = f(1)$ then the function $h(x) = f\{g(x)\}$

• A. has finitely many discontinuities.
• B. is continuous on R.
• C. is discontinuous at some $x = c$.
• D. is a constant function.

Answer 1

Answer: (B)

B

Answer 2

The function g(x) = x – [x] (i.e. the fractional part of x) is discontinuous at integers because it jumps from 1 (as x→n⁻) to 0 (as x→n⁺). However, since f is continuous and f(0) = f(1), the composition h(x) = f(g(x)) “patches” these jumps: at any integer n, the left-hand limit becomes f(1) and the right-hand limit becomes f(0), which are equal. Therefore, h(x) is continuous on ℝ.