# Can we say that all even functions are continuous and non differentiable at x=0 and all odd functions are continuous and differentiable at x=0?

Jitender Singh IIT Delhi
8 years ago
Ans: No
Odd Function:
$f(x) = \frac{1}{x}$
Limitat x =0 does not exist. So f(x) is neither continuous nor differentiable at zero.
$\lim_{h\rightarrow 0^{+}}\frac{1}{x} = \infty$
$\lim_{h\rightarrow 0^{-}}\frac{1}{x} = -\infty$
Even Function:
$f(x) = x^{2}$
To check continuity
$\lim_{h\rightarrow 0^{+}}h^{2} = 0$
$\lim_{h\rightarrow 0^{-}}h^{2} = 0$
To check differentiability
$RHD = \lim_{h\rightarrow 0}\frac{h^{2}-0}{h} = \lim_{h\rightarrow 0}h = 0$
$LHD = \lim_{h\rightarrow 0}\frac{(-h)^{2}-0}{-h} = \lim_{h\rightarrow 0}(-h) = 0$
So f(x) is continuous & differentiable at zero.
Thanks & Regards
Jitender Singh
IIT Delhi