Question

Prove that constant function f(x)=C where C is a constant, is differentiable for each value of x

Answer 1

Let's assume this is a constant function on RR (i.e. f:R→Rf:R→R, f(x)=cf(x)=c for some fixed cc, for all x∈Rx∈R).

Fix any x0x0. What is

limh→0f(x0+h)−f(x0)h  ?limh→0f(x0+h)−f(x0)h  ?

In particular, are there any points where this limit fails to exist? And at the points where the limit does exist, what is the limit equal to?



Thus, if we define a function

f′(x)=limh→0f(x+h)−f(x)h,f′(x)=limh→0f(x+h)−f(x)h,

then f′ is defined for all x, and it is continuous everywhere, as required.