Checking when a real function is continuous in an interval where it is monotonic

Question

I was reviewing for my Real Analysis 1 exam and I found this theorem:

Let $f : I \to \mathbb R$ be a function, where $I\subseteq\mathbb R$ is an interval. Suppose that $f$ is monotonic in $I$, then the following statements are equivalent:

$f$ is continuous in $I$
$f(I)$ is an interval

I am almost finished with this exam but I never used this equivalence in practice, my question is: can you give me some examples where studying the image of an interval is easier or more useful than directly checking the limits of the function, to know when a function is continuous?

Note that the implication "$f$ continuous on $I$ $\implies f(I)$ is an interval" is true without the monotone condition. — J. De Ro, Aug 17 '20 at 16:39
FYI, for real valued functions defined on an interval, the "$f(I)$ is an interval" property is equivalent to the Darboux property (= intermediate value property), which all continuous functions (and all derivatives, even when not continuous) satisfy and which many discontinuous functions also satisfy. — Dave L. Renfro, Aug 17 '20 at 19:02
@DaveL.Renfro a function which satisfies the Darboux property and is monotonic in an interval is automatically continuous in that interval then? — TitorP, Aug 17 '20 at 19:33
Yes. The main idea behind this is that each discontinuity of a monotone function has to be a jump discontinuity (requires careful proof), and each function with a jump discontinuity will not satisfy the intermediate value property (in the vicinity of the domain point at which the jump discontinuity occurs). So if we assume $f$ is monotone, then: "not continuous implies not Darboux", which is equivalent to "Darboux implies continuous". This isn't all you have to prove for that exam question, but it's probably the most difficult part of it. ("Probably", because I don't know what you can use.) — Dave L. Renfro, Aug 17 '20 at 19:50
@DaveL.Renfro I see, that's very interesting. I'll try to study this concepts more in depth. Thanks for your contribution! — TitorP, Aug 17 '20 at 20:09
See also Monotonic function only has jump discontinuities AND Show that one-sided limits always exist for a monotone function (on an interval) AND Proving a bounded monotone function has finite one-sided limits AND Show that for any $x_0\in \mathbb{R}$, the one sided limits exist and that $f^+(x_0)\geq f^-(x_0)$. — Dave L. Renfro, Aug 17 '20 at 20:32

score 0 · Accepted Answer · answered Aug 17 '20 at 16:37

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One way to show that the Cantor function is continuous is by showing that it is non-decreasing and has image $[0,1]$.

answered Aug 17 '20 at 16:37

J. De Ro

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Why the downvote? – J. De Ro Aug 17 '20 at 16:42
Thanks for your answer. So you have to work with weird functions to actually put it to good use, by weird I just mean non-elementary, definitely not polynomials for example. – TitorP Aug 17 '20 at 16:51
I answer to your comment on my original post here: I know, in fact the part that I was curious about was the opposite implication, since the one you are referring to is true for non-monotonic functions too and, correct me if I'm wrong, it is equivalent to the Intermediate value theorem. This makes it, let's say, more usable. – TitorP Aug 17 '20 at 16:55
I agree with what you say above. – J. De Ro Aug 17 '20 at 16:57

Checking when a real function is continuous in an interval where it is monotonic

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