4. Bump functions. Examples of compactly supported functions in S(R) are very handy in many applications in analysis. Some examples are: (a) Suppose a <b, and f is the function such that f(x) = 0 if x < a or x > b and f(x)=e-1/(-a)e-1/(b-x) if a<x<b. Show that f is indefinitely differentiable on R. (b) Prove that there exists an indefinitely differentiable function F on R such that F(x) = 0 if x <a, F(x) = 1 if x > b, and F is strictly increasing on [a, b].