Prove that there is a function g(x) that is not everywehre continuous, yet it is even. Prove that there is a function h(x) that is not everywhere continuous, yet is odd.

Suppose J () is a continuous function. Prove that fe(x) := is an even function and that fo(x) := is an odd function. Prove that there is a function that is neither odd nor even. Prove there is a function that is both odd and even. Prove that f = fe + fo Prove that there is a function 9( ) that is not everywhere continuous, yet is odd.

Suppose f: IR→IR is a monotonic function. True or false: If f is differentiable, then f must be uniformly continuous. True False

Let f be a real-valued continuous and differentiable function. Let function g be defined by g(x) = f(|x| + 2). A student presents the following proof to show that there exists a real number c € (-1, 1) such that g/(c) = 0. (1) Since f is a continuous function, so is g over the interval [-1, 1]. (II) Since f is differentiable, so is g over the interval (-1, 1). (III) It is evident from the definition of g that g(−1) = g(1). (IV) If the above conditions hold, then by Rolle's theorem, there exists dg(x) c = (-1, 1) such that gl (c) dx |x=c Which statement about this proof is correct? = 0. Step (1) does not hold, and hence Rolle's theorem does not apply. Step (II) does not hold, and hence Rolle's theorem does not apply. Step (III) does not hold, and hence Rolle's theorem does not apply. Step (IV) does not hold, and hence the conclusion is false. The proof is completely correct, and the conclusion holds.