True or false: a function f: R → R is continuous at a point a ER if and only if there exists a sequence (x₁) such that Xn a and f(x) → f(a).

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 be a function defined on all of R that satisfies theadditive condition f(x + y) = f(x) + f(y) for all x, y ∈ R (a) Show that f(0) = 0 and that f(−x) = −f(x) for all x ∈ R. (b) Let k = f(1). Show that f(n) = kn for all n ∈ N, and then prove thatf(z) = kz for all z ∈ Z. Now, prove that f(r) = kr for any rationalnumber r.

A function f : R → R is continuous at the point a ∈ R if (and only if)it satisfies the following condition:∀ > 0, ∃δ > 0, |x − a| < δ −→ |f(x) − f(a)| < .(The universe for all variables is R.)Prove that for all a, m, b ∈ R, the function f(x) = mx + b is continuous at a.Remark: A function f : R → R is continuous at the point a ∈ R if (and only if)limx→a f(x) = f(a).