(Sandwich principles.) Let U be an open set in R n , and let a be a point of U.

(a) Let f, g, h: U R be real-valued functions such that f(x) g(x) h(x) for all x in U.

If f(a) = h(a)let's call the common value cand if f and h are continuous at a, prove that g(a) = c and that g is continuous at a as well.

(b) Let f, g, h be real-valued functions defined on U, except possibly at the point a, such that f(x) g(x) h(x) for all x in U, except possibly when x = a. If limxa f(x) = limxa h(x) = L, prove that limxa g(x) = L, too