# Continuous Image of Connected Space is Connected/Corollary 3

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## Corollary to Continuous Image of Connected Space is Connected

Let $\mathbb I = \left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f: \mathbb I \to \R$ be a continuous mapping.

Then $f$ has the intermediate value property.

## Proof

From Subset of Real Numbers is Interval iff Connected, $\mathbb I$ is connected.

By Continuous Image of Connected Space is Connected: corollary 2, $f \left({\mathbb I}\right)$ is also connected.

So from Subset of Real Numbers is Interval iff Connected $f \left({\mathbb I}\right)$ is an real interval.

The result follows by definition of an real interval and the I.V.P.

$\blacksquare$

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $6.2$: Connectedness: Corollary $6.2.14$