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Wikipedia says of the Intermediate Value Theorem that

In mathematical analysis, the intermediate value theorem states that if $f$ is a continuous function whose domain contains the interval $[a, b]$, then it takes on any given value between $f(a)$ and $f(b)$ at some point within the interval.

Of Darboux's Theorem it says

It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval.

What is meant by "the image of an interval is also an interval"?

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Let $f:(X,d_{X})\to(Y,d_{Y})$ be a continuous mapping between metric spaces.

If $C\subseteq X$ is connected, then its image $f(C)\subseteq Y$ is also connected.

In the context of the real line, the connected sets are exactly the intervals.

Consequently, if $X = Y = \mathbb{R}$ and we are given an interval $I\subseteq\mathbb{R}$, $f(I) = J\subseteq\mathbb{R}$ is also connected.

Hence $J$ is an interval, and we are done.

Hopefully this helps!

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  • $\begingroup$ Is there really no simpler way to explain the concept behind the phrase in question? I am studying calculus, but I don't know the definitions of metric space and connected. There must be simple, intuitive way to get at the idea here. $\endgroup$
    – evianpring
    May 20 at 12:26
  • $\begingroup$ @evianpring You can think about connected sets as unbreakable blocks. If the mapping is continuous, it can only deform each block, but it cannot split them into two or more blocks. In the case of the real line, the unbreakable blocks are the intervals. Hence the image of intervals by continuous functions are intervals as well. $\endgroup$
    – Átila Correia
    May 20 at 19:04

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