Let with and let .
(a) Define what it means for to be continuous at .
is said to have a local minimum at if there is some such that whenever and .
If has a local minimum at and is differentiable at , show that .
(b) is said to be convex if
for every and . If is convex, and , prove that
for every .
Deduce that if is convex then is continuous.
If is convex and has a local minimum at , prove that has a global minimum at , i.e., that for every . [Hint: argue by contradiction.] Must be differentiable at ? Justify your answer.