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  • If is an even function such that <img alt="\mathrm{\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}}" src=

If \mathrm{f} is an even function such that \mathrm{\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}} has some finite non-zero value, then

Option: 1

f is continuous and derivable at x=0


Option: 2

f is continuous but not differentiable at x=0


Option: 3

f may be discontinuous at x= 0


Option: 4

None of these


Answers (1)

best_answer

Let \mathrm{f^{\prime}\left(0^{+}\right)=\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}=k (say)}

\mathrm{\therefore \quad f^{\prime}\left(0^{-}\right) =\lim _{h \rightarrow 0} \frac{f(0)-f(0-h)}{h} }
                        \mathrm{=\lim _{h \rightarrow 0} \frac{f(0)-f(h)}{h}=-k . }

Since \mathrm{f^{\prime}\left(0^{+}\right) \neq f^{\prime}\left(0^{-}\right) }, but both are finite, we can say that \mathrm{f(x)}  is continuous at \mathrm{x= 0} but not differentiable at \mathrm{x= 0}.

Posted by

Irshad Anwar

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