If f is an even function such that displaystyle lim h arrow 0 (f(h)-f(0)/h) has some finite non-zero value, then

Question

If f is an even function such that displaystyle lim h arrow 0 (f(h)-f(0)/h) has some finite 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

Tardigrade Admin · Accepted Answer

Let f'(0+)= displaystyle lim h arrow 0 (f(h)-f(0)/h)=k (say) ∴ f'(0-)= displaystyle lim h arrow 0 (f(0)-f(0-h)/h) = displaystyle lim h arrow 0 (f(0)-f(h)/h)=-k Since f'(0+) ≠ f'(0-), but both are finite, we can say that f(x) is continuous at x=0 but not differentiable at x=0

Q. If $f$ is an even function such that $h \to 0 lim \frac{f ( h ) - f ( 0 )}{h}$ has some finite non-zero value, then

$f$ is continuous and derivable at $x = 0$

$f$ is continuous but not differentiable at $x = 0$

$f$ may be discontinuous at $x = 0$

None of these

Q. If f is an even function such that h→0lim​hf(h)−f(0)​ has some finite non-zero value, then

f is continuous and derivable at x=0

f is continuous but not differentiable at x=0

f may be discontinuous at x=0

None of these

Let f′(0+)=h→0lim​hf(h)−f(0)​=k (say) ∴f′(0−)=h→0lim​hf(0)−f(0−h)​ =h→0lim​hf(0)−f(h)​=−k Since f′(0+)=f′(0−), but both are finite, we can say that f(x) is continuous at x=0 but not differentiable at x=0

Q. If $f$ is an even function such that $h \to 0 lim \frac{f ( h ) - f ( 0 )}{h}$ has some finite non-zero value, then

$f$ is continuous and derivable at $x = 0$

$f$ is continuous but not differentiable at $x = 0$

$f$ may be discontinuous at $x = 0$