# Math Help - Examples of bounded/continuous sets and functions

1. ## Examples of bounded/continuous sets and functions

Give an example of a set X and a function f: X -> Real numbers such that:

1) f is continuous at 2 points but is discontinuous everywhere else on X.

I have put this:

f(x)=
x^2-x where x belongs to rationals

0 where x doesn't belong to rationals on X= real numbers

Is this right?

2) f is continuous on X and sup {f(x)|x belongs to X} belongs to {f(x)|x belongs to X} but inf{f(x)|x belongs to X} doesn't belong to {f(x)|x belongs to X}

Could this be a possible answer? :
f(x) = x^2 on X = (-10,10]

3) f is bounded on X but not continuous on X

I'm not sure what to put here.

2. ## Re: Examples of bounded/continuous sets and functions

Originally Posted by CourtneyMoon
Give an example of a set X and a function f: X -> Real numbers such that:
1) f is continuous at 2 points but is discontinuous everywhere else on X.
I have put this:
f(x)=x^2-x where x belongs to rationals
0 where x doesn't belong to rationals on X= real numbers
That will work.
Originally Posted by CourtneyMoon
Give an example of a set X and a function f: X -> Real numbers such that:
2) f is continuous on X and sup {f(x)|x belongs to X} belongs to {f(x)|x belongs to X} but inf{f(x)|x belongs to X} doesn't belong to {f(x)|x belongs to X}
Could this be a possible answer? :
$f(x) = x^2$ on $X = (-10,10]$
The range of your $f(x)$ is $[0,100]$
Does the range contain both if $\inf~\&~\sup~?$
Try $f(x)=-x^2$ on $(-1,1).$

Originally Posted by CourtneyMoon
Give an example of a set X and a function f: X -> Real numbers such that:
3) f is bounded on X but not continuous on X
Let $X=[1,2]$ and $f(x) = \left\{ {\begin{array}{*{20}c} {x,} & {rational} \\ {0,} & {irrational} \\ \end{array} } \right.$.

3. ## Re: Examples of bounded/continuous sets and functions

Originally Posted by Plato
That will work.

The range of your $f(x)$ is $[0,100]$
Does the range contain both if $\inf~\&~\sup~?$
Try $f(x)=-x^2$ on $(-1,1).$

Let $X=[1,2]$ and $f(x) = \left\{ {\begin{array}{*{20}c} {x,} & {rational} \\ {0,} & {irrational} \\ \end{array} } \right.$.
Okay thanks, would f(x)= -x^2 on X =(-1,1) (which you said) work on X=(-1,1] too?
For the f is bounded on X but not continuous on X could I possibly have this:
f(x)= mod x on X = [-10,10] ?

4. ## Re: Examples of bounded/continuous sets and functions

Originally Posted by CourtneyMoon
Okay thanks, would f(x)= -x^2 on X =(-1,1) (which you said) work on X=(-1,1] too?
NO! It does not work for $X=(-1,1]$.
For now the range is $[-1,0]$ which contains the $\inf$.

Originally Posted by CourtneyMoon
For the f is bounded on X but not continuous on X could I possibly have this: f(x)= mod x on X = [-10,10] ?
What does f(x)= mod x mean?

5. ## Re: Examples of bounded/continuous sets and functions

Hmm okay, thank you.

Sorry I meant f(x)=|x| on X = [-10,10]

6. ## Re: Examples of bounded/continuous sets and functions

Originally Posted by CourtneyMoon
Hmm okay, thank you.
Sorry I meant f(x)=|x| on X = [-10,10]
NO again! The function $f(x)=|x|$ is continuous everywhere.
The function I suggested is continuous nowhere on $[1,2]$.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 21, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9610063433647156, "perplexity": 2100.0690839827494}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510266597.23/warc/CC-MAIN-20140728011746-00268-ip-10-146-231-18.ec2.internal.warc.gz"}