Continuity and differentiability of elementary functions

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Given a single-variable elementary function (not piecewise), I was wondering if it is continuous and differentiable in all of its maximum domain? (Not considering the trivial examples involving absolute value function)



Also by piecewise I mean functions that can only be expressed in a piecewise manner. E.g. $cos x$ is not considered piecewise, although it can be expressed in a piecewise manner.



I think the definition of elementary function that I’m going for is the one found on wikipedia: https://en.m.wikipedia.org/wiki/Elementary_function
I think it is also required for the function’s domain to at least contain some interval, but I’m not sure.



Actually I know that if an elementary function is differentiable at some point, its derivative at that point is also given by an elementary function, but I am trying to find out if every elementary function is smooth given that it is defined at some point.



I haven’t been able to find anything useful online, so a rigorous proof would be greatly appreciated. Also please try to keep it as simple as possible as my math is bad.




calculus real-analysis limits continuity elementary-functions




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    up vote
    1
    down vote

    favorite












    Given a single-variable elementary function (not piecewise), I was wondering if it is continuous and differentiable in all of its maximum domain? (Not considering the trivial examples involving absolute value function)



    Also by piecewise I mean functions that can only be expressed in a piecewise manner. E.g. $cos x$ is not considered piecewise, although it can be expressed in a piecewise manner.



    I think the definition of elementary function that I’m going for is the one found on wikipedia: https://en.m.wikipedia.org/wiki/Elementary_function
    I think it is also required for the function’s domain to at least contain some interval, but I’m not sure.



    Actually I know that if an elementary function is differentiable at some point, its derivative at that point is also given by an elementary function, but I am trying to find out if every elementary function is smooth given that it is defined at some point.



    I haven’t been able to find anything useful online, so a rigorous proof would be greatly appreciated. Also please try to keep it as simple as possible as my math is bad.




    calculus real-analysis limits continuity elementary-functions




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Given a single-variable elementary function (not piecewise), I was wondering if it is continuous and differentiable in all of its maximum domain? (Not considering the trivial examples involving absolute value function)



      Also by piecewise I mean functions that can only be expressed in a piecewise manner. E.g. $cos x$ is not considered piecewise, although it can be expressed in a piecewise manner.



      I think the definition of elementary function that I’m going for is the one found on wikipedia: https://en.m.wikipedia.org/wiki/Elementary_function
      I think it is also required for the function’s domain to at least contain some interval, but I’m not sure.



      Actually I know that if an elementary function is differentiable at some point, its derivative at that point is also given by an elementary function, but I am trying to find out if every elementary function is smooth given that it is defined at some point.



      I haven’t been able to find anything useful online, so a rigorous proof would be greatly appreciated. Also please try to keep it as simple as possible as my math is bad.




      calculus real-analysis limits continuity elementary-functions




      share|cite|improve this question











      Given a single-variable elementary function (not piecewise), I was wondering if it is continuous and differentiable in all of its maximum domain? (Not considering the trivial examples involving absolute value function)



      Also by piecewise I mean functions that can only be expressed in a piecewise manner. E.g. $cos x$ is not considered piecewise, although it can be expressed in a piecewise manner.



      I think the definition of elementary function that I’m going for is the one found on wikipedia: https://en.m.wikipedia.org/wiki/Elementary_function
      I think it is also required for the function’s domain to at least contain some interval, but I’m not sure.



      Actually I know that if an elementary function is differentiable at some point, its derivative at that point is also given by an elementary function, but I am trying to find out if every elementary function is smooth given that it is defined at some point.



      I haven’t been able to find anything useful online, so a rigorous proof would be greatly appreciated. Also please try to keep it as simple as possible as my math is bad.





      calculus real-analysis limits continuity elementary-functions





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      share|cite|improve this question




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      asked Jul 26 at 16:30









      user63858

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          Elementary functions, as usually defined, are continuos in all the domain of definition but not always differentiable. Let consider as an example of elementary functions continuous but not differentiable at a point



          • $f(x)=sqrt x$ at $x=0$

          • $f(x)=sqrt[3] x$ at $x=0$

          • $f(x)=arcsin x$ at $x=1$





          share|cite|improve this answer























          • Thanks, I should have looked at these examples first, I feel dumb now
            – user63858
            Jul 26 at 16:47










          • It’s ok, now your overview is more clear I hope. You are welcome! Bye
            – gimusi
            Jul 26 at 16:49










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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

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          active

          oldest

          votes








          up vote
          1
          down vote



          accepted










          Elementary functions, as usually defined, are continuos in all the domain of definition but not always differentiable. Let consider as an example of elementary functions continuous but not differentiable at a point



          • $f(x)=sqrt x$ at $x=0$

          • $f(x)=sqrt[3] x$ at $x=0$

          • $f(x)=arcsin x$ at $x=1$





          share|cite|improve this answer























          • Thanks, I should have looked at these examples first, I feel dumb now
            – user63858
            Jul 26 at 16:47










          • It’s ok, now your overview is more clear I hope. You are welcome! Bye
            – gimusi
            Jul 26 at 16:49














          up vote
          1
          down vote



          accepted










          Elementary functions, as usually defined, are continuos in all the domain of definition but not always differentiable. Let consider as an example of elementary functions continuous but not differentiable at a point



          • $f(x)=sqrt x$ at $x=0$

          • $f(x)=sqrt[3] x$ at $x=0$

          • $f(x)=arcsin x$ at $x=1$





          share|cite|improve this answer























          • Thanks, I should have looked at these examples first, I feel dumb now
            – user63858
            Jul 26 at 16:47










          • It’s ok, now your overview is more clear I hope. You are welcome! Bye
            – gimusi
            Jul 26 at 16:49












          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          Elementary functions, as usually defined, are continuos in all the domain of definition but not always differentiable. Let consider as an example of elementary functions continuous but not differentiable at a point



          • $f(x)=sqrt x$ at $x=0$

          • $f(x)=sqrt[3] x$ at $x=0$

          • $f(x)=arcsin x$ at $x=1$





          share|cite|improve this answer















          Elementary functions, as usually defined, are continuos in all the domain of definition but not always differentiable. Let consider as an example of elementary functions continuous but not differentiable at a point



          • $f(x)=sqrt x$ at $x=0$

          • $f(x)=sqrt[3] x$ at $x=0$

          • $f(x)=arcsin x$ at $x=1$






          share|cite|improve this answer















          share|cite|improve this answer



          share|cite|improve this answer








          edited Jul 26 at 16:40


























          answered Jul 26 at 16:35









          gimusi

          65k73583




          65k73583











          • Thanks, I should have looked at these examples first, I feel dumb now
            – user63858
            Jul 26 at 16:47










          • It’s ok, now your overview is more clear I hope. You are welcome! Bye
            – gimusi
            Jul 26 at 16:49
















          • Thanks, I should have looked at these examples first, I feel dumb now
            – user63858
            Jul 26 at 16:47










          • It’s ok, now your overview is more clear I hope. You are welcome! Bye
            – gimusi
            Jul 26 at 16:49















          Thanks, I should have looked at these examples first, I feel dumb now
          – user63858
          Jul 26 at 16:47




          Thanks, I should have looked at these examples first, I feel dumb now
          – user63858
          Jul 26 at 16:47












          It’s ok, now your overview is more clear I hope. You are welcome! Bye
          – gimusi
          Jul 26 at 16:49




          It’s ok, now your overview is more clear I hope. You are welcome! Bye
          – gimusi
          Jul 26 at 16:49












           

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