Question

Let $f:R\to R$ be a function such that $\left|f\left(x\right)\right|\le x^2,$ for all $xϵR$. Then, at $x=0,f$ is :

• A. continuous but not differentiable
• B. continuous as well as differentiable
• C. neither continuous nor differentiable
• D. differentiable but not continuous.

Answer 1

Answer: (B)

$|f(x)|\le x^2$
$|f(0)|\le 0$
$f(0)=0$
$\underset{x\to 0}{\lim }\left|f\left(x\right)\right|\le \underset{x\to 0}{\lim }{x}^2$
$\le 0$
$=0$
Conti at $x=0$
$LHD=\underset{h\to 0}{\lim }\frac{f(-h)-f(0)}{-h}$
$\underset{h\to 0}{\lim }\frac{h^2-0}{-h}=0$
$RHD=\underset{h\to 0}{\lim }\frac{f(h)-f(0)}{h}$
$\underset{h\to 0}{\lim }\frac{h^2-0}{h}=0$
$L.H.D.=R.H.D.$