Basic Analysis I — Real Analysis · Problem 3.3.15

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · extreme-and-intermediate-value-theorems

Problem

Suppose

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

is continuous. a) Prove that if there is a

c

such that

f(c)f(-c) < 0

, then there is a

d \in \mathbb{R}

such that

f(d) = 0

. b) Find a continuous function

f

such that

f(\mathbb{R}) = \mathbb{R}

, but

f(x)f(-x) \geq 0

for all

x \in \mathbb{R}