Define a function g by
(f(x)−f(c))/(x−c) for x≠c
and f(x)ᐁx｢c｣ at x=c.
｢g(x) Ƚx￫c｣
= ｢(f(x)−f(c))/(x−c) Ƚx￫c｣
┆f is differentiable at c┆
= f(x)ᐁx｢c｣
So g is continuous at c.
g(x) = (f(x)−f(c))/(x−c) for x≠c
f(x) = f(c)+g(x)·(x−c) for all x
f(c), g(x) and x−c are all continuous at c.
By sum and product rule of continuity, f is continuous at c.
