1. Let f : R -> R satisfy
f(x + y) = f(x) * f(y)
for all x, y elements of Reals. Prove that if f is differentiable at 0 then f is differen-
tiable on R and
f'(x) = f'(0)f(x):
More generally if f is differentiable at p then f is differentiable on R
and
f'(x) = f'(p)f(x – p).

2. Let I subset Reals be an interval, f : I -> R a function, and p is an element in I.
Prove that f is differentiable at p if and only if there exists a function
phi : I -> R that is continuous at p such that
f(x) = f(p) + (x – p)phi(x).
Moreover, if there exists a function phi: I -> R that is continuous at p
such that
f(x) = f(p) + (x – p)phi(x);
then f'(p) = phi(p).

3. Let I subset Reals be an interval, f : I -> R a function, and p element I. Prove
that if f is differentiable at p then f is continuous at p.