Basic Analysis I — Real Analysis · Problem 3.3.4

Continuous Functions — Jiri Lebl

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

Problem

Let

f(x) := \begin{cases} \sin(1/x) & \text{if } x \neq 0, \\ 0 & \text{if } x = 0. \end{cases}

Show that

f

has the intermediate value property. That is, whenever

a < b

, if there exists a

y

such that

f(a) < y < f(b)

f(a) > y > f(b)

, then there exists a

c \in (a, b)

such that

f(c) = y