1. Let X R be a compact rectangle and let [a, b] R be a compact interval. Let U R+1 be an open set such that XX [a, b] U and let f: U R be a continuously differentiable function on U. Define the function g: [a, b] R by: for each y = [a, b]: g(y) = [f(x, y) dx a. Show that for any yo [a, b] and h E R with yo + h = [a, b], one has: ryoth g(yo + h) g(yo) - h(x, y) dx = af = f(x, yo)) dx dy yo b. Show that for any yo [a, b], the function g is differentiable at yo with derivative given by: af g'(x) = r (x, y) The result is sometimes stated simply as: d (f(x, y) dx) \ - yo dy y=yo ) dx = (x,yo) dx Note that the integral on the right makes sense: our hypotheses ensures that the partial derivative function is a continuous function of (x, y) in U; the integrand on the right ay' is the function x E X obtained by fixing the second variable y to be the value yo, so it is continuous and hence Riemann integrable over X. You may use without proof the following result from real analysis: Theorem: A continuous function op: S R on a compact subset S of Euclidean space (e.g. a compact rectangle) is uniformly continuous; more precisely: for any e R>0, there exists 8 R>0 such that for any 51, 52 ES with 51 - 52| < 8, one has: |4 (51) - (52)|