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.


$\blacksquare$


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