Show that f is differentiable (on R) if only if f´(0) exists?
f:R-->R satisfy the functional equation
f(x+y)=f(x)f(y) (for all x and y in R)
Show that f is differentiable (on R) if only if f´(0) exists
Thanks for your help
More Questions From This User See All
Answers & Comments
Verified answer
If f is differentiable on R, then in particular f'(0) exists. Conversely, if f'(0) exists, then
lim(h -->0) [f(x + h) - f(x)]/h
= lim(h-->0) [f(x)f(h) - f(x)]/h
= lim(h-->0) f(x) (f(h) - 1)/h
= f(x) lim(h-->0) (f(h) - 1)/h
= f(x) f'(0),
for all x in R. Hence f is differentiable on R, with f'(x) = f'(0) f(x) for all x in R.