Lebl-Basic-Analysis-I · Problem 3.5.9

Continuous Functions

Mathematics · limits-at-infinity

Problem

Suppose

f : \mathbb{R} \to \mathbb{R}

is a 2-periodic function, that is

f(x+2) = f(x)

for all

x

. Define

g : \mathbb{R} \to \mathbb{R}

g(x) := f\!\left(\frac{\sqrt{x^2+1}-1}{x}\right)

a) Find the function

\varphi : (-1,1) \to \mathbb{R}

such that

g\bigl(\varphi(t)\bigr) = f(t)

, that is

\varphi^{-1}(x) = \dfrac{\sqrt{x^2+1}-1}{x}

. b) Show that

f

is continuous if and only if

g

is continuous and

\lim_{x \to \infty} g(x) = \lim_{x \to -\infty} g(x) = f(1) = f(-1).