Question

Let $f:R\to R$ and $g:R\to R$ be defined by $g\left(x\right)=xf\left(x\right)$ Then

• A. $g$ is a differentiable function
• B. $g$ is differentiable at 0 if $f$ is continuous at 0
• C. $g$ is one-one if $f$ is one-one
• D. $f$ is a differentiable function

Answer 1

Answer: (B)

$g$ is differentiable at 0 if $f$ is continuous at 0

$g^{\prime }\left(x\right)=f\left(x\right)+xf\left(x\right)$
${lim}_{x\to 0}{g}^{\prime }\left(x\right)=f\left(0^{-}\right)$
${lim}_{x\to 0}{g}^{\prime }\left(x\right)=f\left(0^{+}\right)$
for g'(x) to be differentiable
$f\left(0^{-}\right)=f\left(0^{+}\right)$ which mean f(x) has to be continuous at $x=0$