Continuous Image of Connected Space is Connected/Corollary 2

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Corollary to Continuous Image of Connected Space is Connected

Let $T$ be a connected topological space.

Let $f: T \to \R$ be a continuous real-valued mapping.


Then $f \left({T}\right)$ is a real interval.


Proof

From Continuous Image of Connected Space is Connected, the continuous image of the connected space $T$ is connected.

The result follows from Subset of Real Numbers is Interval iff Connected.

$\blacksquare$


Sources