# A continuous function on an open interval bounded?

I have being taught that a continuous function on a bounded interval is bounded then I am thinking of a counterexample when the interval is open. The example I came up with was $$f(x)=log(x)$$ for $$x \in (0,1)$$, but I am kind of thinking how could I prove its continuity on $$(0,1)$$ from definition?

• Another example is $f(x)=\frac1x$. You have been taught something wrong, or you do not remember it correctly. A continuous function on a closed bounded interval is bounded. – Hagen von Eitzen Jun 12 '19 at 20:05
• To address the second concern, the usual definition of $\log$ is as the inverse function of $\exp$ and the inverse of a monotonic function on an interval is continuous. – Thorgott Jun 12 '19 at 20:12
• Here's an example of demonstration that $log(x)$ is continuous (on $(0, +\infty)$) using the definition. – lurker Jun 12 '19 at 22:05
• You've better have been taught that the image of a compact set unter a continuous function is compact as well. – Michael Hoppe Jun 13 '19 at 10:38
• @MichaelHoppe Ah our course hasn't properly defined what compactness really means, so I am guessing that is no use for now :( (what is compactness anyways?) – JustWandering Jun 13 '19 at 11:59

## 1 Answer

If $$f$$ is continous in a interval [a,b], and $$I\subset[a,b]$$ is an interval, then $$f|_I$$ is continous.

• True, and yesterday it was a rainy night in Georgia. – Michael Hoppe Jun 13 '19 at 10:37
The text passage is a discussion about the continuity of functions on open intervals, with a specific example of \( f(x) = \log(x) \) for \( x \in (0,1) \). It includes comments suggesting alternative examples and methods to prove continuity.
no_question

the passage does not contain a direct mathematical question that requires a numerical or symbolic answer. instead, it is a discussion with comments and suggestions.
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
# A continuous function on an open interval bounded?

I have being taught that a continuous function on a bounded interval is bounded then I am thinking of a counterexample when the interval is open. The example I came up with was $$f(x)=log(x)$$ for $$x \in (0,1)$$, but I am kind of thinking how could I prove its continuity on $$(0,1)$$ from definition?

• Another example is $f(x)=\frac1x$. You have been taught something wrong, or you do not remember it correctly. A continuous function on a closed bounded interval is bounded. – Hagen von Eitzen Jun 12 '19 at 20:05
• To address the second concern, the usual definition of $\log$ is as the inverse function of $\exp$ and the inverse of a monotonic function on an interval is continuous. – Thorgott Jun 12 '19 at 20:12
• Here's an example of demonstration that $log(x)$ is continuous (on $(0, +\infty)$) using the definition. – lurker Jun 12 '19 at 22:05
• You've better have been taught that the image of a compact set unter a continuous function is compact as well. – Michael Hoppe Jun 13 '19 at 10:38
• @MichaelHoppe Ah our course hasn't properly defined what compactness really means, so I am guessing that is no use for now :( (what is compactness anyways?) – JustWandering Jun 13 '19 at 11:59

## 1 Answer

If $$f$$ is continous in a interval [a,b], and $$I\subset[a,b]$$ is an interval, then $$f|_I$$ is continous.

• True, and yesterday it was a rainy night in Georgia. – Michael Hoppe Jun 13 '19 at 10:37
user
The text passage is a discussion about the continuity of functions on open intervals, with a specific example of \( f(x) = \log(x) \) for \( x \in (0,1) \). It includes comments suggesting alternative examples and methods to prove continuity.

TYPE: NO_QUESTION

The passage does not contain a direct mathematical question that requires a numerical or symbolic answer. Instead, it is a discussion with comments and suggestions.