Continuous Image of Connected Space is Connected/Corollary 3

Corollary to Continuous Image of Connected Space is Connected

Let $I = \closedint a b$ be a closed real interval.

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

Then $f$ has the intermediate value property.

Proof

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

By Continuous Image of Connected Space is Connected: Corollary 2, $f \sqbrk I$ is also connected.

So from Subset of Real Numbers is Interval iff Connected, $f \sqbrk I$ is a real interval.

The result follows by definition of an real interval and the intermediate value property.

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Sources

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