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.