1- prove that if y=f(x) is continuous and f(x)=0 when  x is a rational number, then f(x)=0 for all x∈R   2- let f(x)=0 if x∈Q if x and f(x)=1 if x∈R/Q. prove that y=f(x) is not continuous at any point a∈R

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.

Let f(x) = |x| where xeR. Discuss the continuity and differentiability of f(x) at x = 0