McGill University Department of Mathematics and Statistics MATH 243 Analysis 2, Winter 2023 Assignment 5: Solutions 􏰀1 ifx∈{n1:n∈N} Prove that f is Riemann integrable on [0,1] and compute 0 otherwise Hint: You may use either the squeeze theorem or the ε-criterion for Riemann integrability. Either way, this is a challenging question! 1. Solution: Using the

McGill University Department of Mathematics and Statistics MATH 243 Analysis 2, Winter 2023 Assignment 2: Solutions 1. Use the intermediate value theorem and Rolle’s theorem to prove that the function f : R+ → R, f(x) := x3 +2×2 −5+ x1 has exactly 2 roots in R+. You are not permitted to use any Calculator-

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0 if x = 0 McGill University Department of Mathematics and Statistics MATH 243 Analysis 2, Winter 2023 Assignment 6: Solutions f(x) := Prove that f is Riemann integrable on [0, 1]. Then α is constant on [0,4ε] except (potentially) at the right endpoint; thus α is Riemann integrable on this interval. Furthermore, α is

McGill University Department of Mathematics and Statistics MATH 243 Analysis 2, Winter 2023 Assignment 1: Solutions 1. Let f : R → R, f(x) := x1/3. Prove that f is not differentiable at 0. Hint: Considerg◦f whereg:R→R,g(x):=x3. 1. Solution: (Using the hint) Note that g is differentiable on R with g′(0) = 0. Assume that

McGill University Department of Mathematics and Statistics MATH 243 Analysis 2, Winter 2023 Assignment 4: Solutions 1. (a) Let f and g be Riemann integrable on [a,b] such that f(x) ≤ g(x) for all x ∈ [a,b]. Prove 􏰂b 􏰂b that f ≤ g. aa (b) Let f be Riemann integrable on [a,b] and let