# Trying to prove that if $f:[a, b]\to[s, t]$ is monotone then $f$ is continuous

I'm trying to prove that if $f:[a, b]\to[s, t]$ is monotone (and its image is closed interval) then $f$ is continuous.

My attempt:

I say wlog, $f$ is increasing. I know that a monotone function only has jump discontinuities, so let's say $c\in [a, b]$ is a jump discontinuity, so:

$$K=\lim\limits_{x\to c^-} f(x)<\lim\limits_{x\to c^+} f(x)=L$$

So for each $c<x_n\to c$, $f(x_n)\to L$. $x_n>c$ so $f(x_n)\ge f(c)$ so $f(c)\le L$.

And for each $c>y_n\to c$, $f(y_n)\to K$. $y_n<c$ so $f(y_n)\le f(c)$ so $f(c)\ge K$.

So $L=K$ and this is the contradiction.

And my question - where did I use that the image of $f$ is closed interval? Did I do something wrong?

thanks?

• Your conclusion that $L=K$ is not supported by what you have done so far. – Alex S Jul 7 '15 at 22:03
• Oh I see.... Is this the right way though? – Stabilo Jul 7 '15 at 22:05
• As you noticed, you need to incorporate the fact that the function's image is an interval. So there must be points in $[a,b]$ which map to points between $L$ and $K$, but how? – Alex S Jul 7 '15 at 22:07
• Using my attempt of solution you mean, or another way? – Stabilo Jul 7 '15 at 22:09
• After you conclude that $K\leq f(c)\leq L$, show that the existence of $K<y<f(c)$ or $f(c)<y<L$ leads to a contradiction. Conclude that $K=f(c)=L$. – Alex S Jul 7 '15 at 22:12
The text passage is a discussion about a proof attempt for a theorem in real analysis. The author is asking for feedback on their proof and specifically inquires about the use of the condition that the image of the function is a closed interval.
single_question

the passage contains a single mathematical question asking where the proof uses the fact that the image of the function is a closed interval. no answer to this question is provided in the text, only suggestions for how to proceed.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Trying to prove that if $f:[a, b]\to[s, t]$ is monotone then $f$ is continuous

I'm trying to prove that if $f:[a, b]\to[s, t]$ is monotone (and its image is closed interval) then $f$ is continuous.

My attempt:

I say wlog, $f$ is increasing. I know that a monotone function only has jump discontinuities, so let's say $c\in [a, b]$ is a jump discontinuity, so:

$$K=\lim\limits_{x\to c^-} f(x)<\lim\limits_{x\to c^+} f(x)=L$$

So for each $c<x_n\to c$, $f(x_n)\to L$. $x_n>c$ so $f(x_n)\ge f(c)$ so $f(c)\le L$.

And for each $c>y_n\to c$, $f(y_n)\to K$. $y_n<c$ so $f(y_n)\le f(c)$ so $f(c)\ge K$.

So $L=K$ and this is the contradiction.

And my question - where did I use that the image of $f$ is closed interval? Did I do something wrong?

thanks?

• Your conclusion that $L=K$ is not supported by what you have done so far. – Alex S Jul 7 '15 at 22:03
• Oh I see.... Is this the right way though? – Stabilo Jul 7 '15 at 22:05
• As you noticed, you need to incorporate the fact that the function's image is an interval. So there must be points in $[a,b]$ which map to points between $L$ and $K$, but how? – Alex S Jul 7 '15 at 22:07
• Using my attempt of solution you mean, or another way? – Stabilo Jul 7 '15 at 22:09
• After you conclude that $K\leq f(c)\leq L$, show that the existence of $K<y<f(c)$ or $f(c)<y<L$ leads to a contradiction. Conclude that $K=f(c)=L$. – Alex S Jul 7 '15 at 22:12
user
The text passage is a discussion about a proof attempt for a theorem in real analysis. The author is asking for feedback on their proof and specifically inquires about the use of the condition that the image of the function is a closed interval.

TYPE: SINGLE_QUESTION

The passage contains a single mathematical question asking where the proof uses the fact that the image of the function is a closed interval. No answer to this question is provided in the text, only suggestions for how to proceed.