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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?

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?

Grade:9

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
7 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
askIITians Faculty

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