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If is an even function such that lim(h r...

If is an even function such that `lim_(h rarr 0) (f(h)-f(0))/(h)` has some fininte non-zero value, then

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

Text Solution

Verified by Experts

The correct Answer is:
B
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.
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