If `f` is an even function such that `(lim)_(hvec0)(f(h)-f(0))/h` has some finite non-zero value, then prove that `f(x)` is not differentiable at `x=0.`

Options

A. f is continuous and derivable at x =0
B. f is continuous but not differentiable at x = 0
C. f may be discontinuous at x = 0
D. None of these

Answer

B

Explanation

Let `f'(0^(+))=underset(hrarr0)(lim)(f(h)-f(0))/(h)=k(say)`
`therefore" "f'(0^(-))=underset(hrarr0)(lim)(f(0)-f(0-h))/(h)`
`" "=underset(hrarr0)(lim)(f(0)-f(h))/(h)=-k.`
Since `f'(0^(+))ne f'(0^(-))`, but both are finite, we can say that f(x) is continuous at x = 0 but not differentiable at x = 0.