Let 
and define 
.
a) Must 
be differentiable at 0?
b) Must have any differentiable points?
c) Let 
, show that 
exists and determine what it is.
Solutions.
a) No. For example, let 
for 
so that 
near zero. This 
but 
does not exist.
b) Yes. In fact, must be differentiable almost everywhere, by the Lebesgue theorem on the differentiation of the integral. This theorem requires only that 
which is true.
c) By the Schwarz inequality,
At least that’s it for 
Being careful about the other side, we determine that
But since 
is an integrable function we have (by an argument that uses the Dominated Convergence Theorem) that
Hence 
so 
be a metric space and 
a function with 
for 
. Prove that 
is a metric on 
.
(because f is increasing), so 
. Of course, 
for 
(because f is increasing) and 
because 
(because f is increasing) 
(because f(0) = 0 and f is concave down), so triangle inequality is verified.
(then you can substitute a = d(x,y), b = d(y,z))
, 
. Sum them up, you are done.
