Lebl-Basic-Analysis-I · Problem 7.5.6

Metric Spaces

Mathematics · continuous-functions

Problem

Prove the following version of the intermediate value theorem. Let

(X, d)

be a connected metric space and

f \colon X \to \mathbb{R}

a continuous function. Suppose

x_0, x_1 \in X

and

y \in \mathbb{R}

are such that

f(x_0) < y < f(x_1)

. Then prove that there exists a

z \in X

such that

f(z) = y

. Hint: See Exercise 7.5.5.