Pointwise product of uniformly continuous functions












2














True or false: Let $f(x)$ and $g(x)$ be uniformly continuous functions from $mathbb{R}$ to $mathbb{R}$.
Then their pointwise product $f(x)g(x)$ is uniformly continuous.



I think it will be true; take $f(x)= g(x) = sqrt x$.



Am I right or wrong?










share|cite|improve this question




















  • 7




    You don't prove a statement true by finding an example for which the statement holds.
    – ab123
    Nov 11 at 12:52










  • It would be true if you added "bounded" to your assumptions, but as stated JCSantos has the perfect counterexample in his answer.
    – JonathanZ
    Nov 11 at 17:57


















2














True or false: Let $f(x)$ and $g(x)$ be uniformly continuous functions from $mathbb{R}$ to $mathbb{R}$.
Then their pointwise product $f(x)g(x)$ is uniformly continuous.



I think it will be true; take $f(x)= g(x) = sqrt x$.



Am I right or wrong?










share|cite|improve this question




















  • 7




    You don't prove a statement true by finding an example for which the statement holds.
    – ab123
    Nov 11 at 12:52










  • It would be true if you added "bounded" to your assumptions, but as stated JCSantos has the perfect counterexample in his answer.
    – JonathanZ
    Nov 11 at 17:57
















2












2








2







True or false: Let $f(x)$ and $g(x)$ be uniformly continuous functions from $mathbb{R}$ to $mathbb{R}$.
Then their pointwise product $f(x)g(x)$ is uniformly continuous.



I think it will be true; take $f(x)= g(x) = sqrt x$.



Am I right or wrong?










share|cite|improve this question















True or false: Let $f(x)$ and $g(x)$ be uniformly continuous functions from $mathbb{R}$ to $mathbb{R}$.
Then their pointwise product $f(x)g(x)$ is uniformly continuous.



I think it will be true; take $f(x)= g(x) = sqrt x$.



Am I right or wrong?







real-analysis uniform-continuity






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 11 at 19:04









jwodder

1,262918




1,262918










asked Nov 11 at 12:50









santosh

969




969








  • 7




    You don't prove a statement true by finding an example for which the statement holds.
    – ab123
    Nov 11 at 12:52










  • It would be true if you added "bounded" to your assumptions, but as stated JCSantos has the perfect counterexample in his answer.
    – JonathanZ
    Nov 11 at 17:57
















  • 7




    You don't prove a statement true by finding an example for which the statement holds.
    – ab123
    Nov 11 at 12:52










  • It would be true if you added "bounded" to your assumptions, but as stated JCSantos has the perfect counterexample in his answer.
    – JonathanZ
    Nov 11 at 17:57










7




7




You don't prove a statement true by finding an example for which the statement holds.
– ab123
Nov 11 at 12:52




You don't prove a statement true by finding an example for which the statement holds.
– ab123
Nov 11 at 12:52












It would be true if you added "bounded" to your assumptions, but as stated JCSantos has the perfect counterexample in his answer.
– JonathanZ
Nov 11 at 17:57






It would be true if you added "bounded" to your assumptions, but as stated JCSantos has the perfect counterexample in his answer.
– JonathanZ
Nov 11 at 17:57












2 Answers
2






active

oldest

votes


















5














It is false. Just take $f(x)=g(x)=x$.






share|cite|improve this answer





























    1














    The uniform continuity property refers to a spatially homogeneous behavior of continuity. Look at the graph of the following functions.




    1. $f(x)=x$

    2. $f(x)=sqrt{|x|}$

    3. $f(x)=x^2$

    4. $$ f(x)= begin{cases}
      xsin(frac{1}{x}) & xne 0 \
      0 & x=0
      end{cases}$$


    Note that (2) has a "vertiginous behavior" around $0$, since that $$f'(x)=frac{1}{2sqrt{x}}to infty$$ when $xdownarrow 0$.
    Also for (3) this behavior occurs when $|x|to infty$.
    (4) is continuous but is even "less uniform" than previous cases.






    share|cite|improve this answer





















      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2993834%2fpointwise-product-of-uniformly-continuous-functions%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      5














      It is false. Just take $f(x)=g(x)=x$.






      share|cite|improve this answer


























        5














        It is false. Just take $f(x)=g(x)=x$.






        share|cite|improve this answer
























          5












          5








          5






          It is false. Just take $f(x)=g(x)=x$.






          share|cite|improve this answer












          It is false. Just take $f(x)=g(x)=x$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 11 at 12:53









          José Carlos Santos

          149k22117219




          149k22117219























              1














              The uniform continuity property refers to a spatially homogeneous behavior of continuity. Look at the graph of the following functions.




              1. $f(x)=x$

              2. $f(x)=sqrt{|x|}$

              3. $f(x)=x^2$

              4. $$ f(x)= begin{cases}
                xsin(frac{1}{x}) & xne 0 \
                0 & x=0
                end{cases}$$


              Note that (2) has a "vertiginous behavior" around $0$, since that $$f'(x)=frac{1}{2sqrt{x}}to infty$$ when $xdownarrow 0$.
              Also for (3) this behavior occurs when $|x|to infty$.
              (4) is continuous but is even "less uniform" than previous cases.






              share|cite|improve this answer


























                1














                The uniform continuity property refers to a spatially homogeneous behavior of continuity. Look at the graph of the following functions.




                1. $f(x)=x$

                2. $f(x)=sqrt{|x|}$

                3. $f(x)=x^2$

                4. $$ f(x)= begin{cases}
                  xsin(frac{1}{x}) & xne 0 \
                  0 & x=0
                  end{cases}$$


                Note that (2) has a "vertiginous behavior" around $0$, since that $$f'(x)=frac{1}{2sqrt{x}}to infty$$ when $xdownarrow 0$.
                Also for (3) this behavior occurs when $|x|to infty$.
                (4) is continuous but is even "less uniform" than previous cases.






                share|cite|improve this answer
























                  1












                  1








                  1






                  The uniform continuity property refers to a spatially homogeneous behavior of continuity. Look at the graph of the following functions.




                  1. $f(x)=x$

                  2. $f(x)=sqrt{|x|}$

                  3. $f(x)=x^2$

                  4. $$ f(x)= begin{cases}
                    xsin(frac{1}{x}) & xne 0 \
                    0 & x=0
                    end{cases}$$


                  Note that (2) has a "vertiginous behavior" around $0$, since that $$f'(x)=frac{1}{2sqrt{x}}to infty$$ when $xdownarrow 0$.
                  Also for (3) this behavior occurs when $|x|to infty$.
                  (4) is continuous but is even "less uniform" than previous cases.






                  share|cite|improve this answer












                  The uniform continuity property refers to a spatially homogeneous behavior of continuity. Look at the graph of the following functions.




                  1. $f(x)=x$

                  2. $f(x)=sqrt{|x|}$

                  3. $f(x)=x^2$

                  4. $$ f(x)= begin{cases}
                    xsin(frac{1}{x}) & xne 0 \
                    0 & x=0
                    end{cases}$$


                  Note that (2) has a "vertiginous behavior" around $0$, since that $$f'(x)=frac{1}{2sqrt{x}}to infty$$ when $xdownarrow 0$.
                  Also for (3) this behavior occurs when $|x|to infty$.
                  (4) is continuous but is even "less uniform" than previous cases.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 11 at 13:38









                  Daniel Camarena Perez

                  57538




                  57538






























                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.





                      Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                      Please pay close attention to the following guidance:


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2993834%2fpointwise-product-of-uniformly-continuous-functions%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      這個網誌中的熱門文章

                      Xamarin.form Move up view when keyboard appear

                      Post-Redirect-Get with Spring WebFlux and Thymeleaf

                      Anylogic : not able to use stopDelay()