a) Prove that for every e such that 0 < € < M/2 there is a non-empty open interval I≤ [a, b] such that cb (M - €)P|I| ≤ * \ƒ(x)\Pdx ≤ M²(b − a). b) Deduce that cb 1/p Him: (/* IF(x)/'da) ""- \ƒ(x)\³dx) = P→∞ = M. =

7. Modify B(z) to construct a C function C(z) which satisfies a. C(z) = 0 if S 0.. b. C(z) = 1 if > 1. c. C'(z) > 0 if0< z < 1.

3. Define f(x) so that 16) = { exp (교1), |피 1 f(r) = 0, (a) Explain why the function f(x) must be continuous everywhere except perhaps at x = +1. Hint: How is f(x) constructed from known continuous functions? (b) State the definition for f(x) to be continuous at c. Assume that c is not an endpoint. (c) Prove that f(x) is in fact continuous at both –1 and 1. (d) State the limit definition for f(x) to be differentiable at c. Assume that c is not an endpoint. (e) Explain why the function f(x) must be differentiable everywhere except perhaps at r = ±1. Hint: How is f(x) constructed from known differentiable functions? (f) Prove that f(x) is in fact differentiable at both -1 and 1.

4. Sketch the graph of an example function f on domain [0, 10] with jump discontinuities at x = 3 (righ continuous), x = 5 (left continuous), and x = 7 (neither left nor right continuous), where f(3) = 3, f (5) = 5, f (7) = 7.