Question

Let $f : R \to R$ be a function such that $\left|f\left(x\right)\right| \le x^{2},$ for all $x \epsilon R$. 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 )| \leq\, x ^{2} $
$| f (0)| \leq \,0 $
$f (0)=0$
$\displaystyle\lim_{x\to0} \left|f \left(x\right)\right| \le\, \displaystyle\lim_{x\to0} \, x^{2}$
$ \leq\, 0 $
$=0$
Conti at $ x =0$
$LHD =\displaystyle\lim_{h \to 0} \frac{ f (- h )- f (0)}{- h }$
$\displaystyle\lim_{h \to 0} \frac{h^{2}-0}{-h}=0 $
$RHD = \displaystyle\lim_{h \to 0} \frac{f(h)-f(0)}{h} $
$\displaystyle\lim_{h \to 0} \frac{h^{2}-0}{h}=0 $
$ L.H.D. = R.H.D. $