Question

Let $f:R\to R$ be any function. Define $g:R\to R$ by $g\left(x\right)=|f\left(x\right)|$for all $x$. Then $g$ is

• A. g may be bounded even if f is unbounded
• B. one-one if f is one
• C. continuous if f is continuous
• D. differentiable if f is differentiable

Answer 1

The correct option is C continuous if f is continuous

take $f\left(x\right)=x,$

Since $g:R\to R$ given by $g\left(x\right)=\left|x\right|$ is not one-one.

So option (B) is violated.

Also $g$ is not differetiable at $x=0$

Let $u\left(x\right)=\left|x\right|$ then $u$ is continuous function and $g=u\left(f\left(x\right)\right)=u\cdot f\left(x\right)$

Hence $g$ is continuous if $f$ is continuous.

It is easy to see $g$ is bounded if and only if $f$ is bounded.