Give an example of a continuous strictly increasing function g:R->R which is...

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Solution 1

Here is a way to construct such a function.

  1. Draw $f(x)=x^3+x^2+x$.
  2. At every integer $n$, draw $g(x)$ between $n$ and $n+1$ as the secant from $f(n)$ to $f(n+1)$.
  3. $g(x)$ is now continuous, differentiable at every point between the integers, but not at the integers.

EDIT v2 I've edited to use a strictly increasing function.


Here is a graphical depiction of the concept:

enter image description here

Solution 2

Yes, a piecewise defined function will do it. We first do it for $x\ge 0$.

For $0\le x\lt 1$, let $f(x)=x$. For $1\le x\lt 2$, let $f(x)=1+2x$. For $2\le x\lt 3$, let $f(x)=3+4x$. For $3\le x\lt 4$, let $f(x)=7+8x$, and so on.

So for $0\le n\le x\lt n+1$, let $f(x)=2^n-1+2^n x$.

Verification that we do not have differentiability at the integers is reasonably straightforward, the slopes do not match.

For the part with $x\lt 0$, use $-(f(|x|+1)+1$.

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Updated on February 20, 2020

Comments

  • user108626
    user108626 over 3 years

    Give an example of a continuous strictly increasing function $g\colon \Bbb R\to \Bbb R$ which is differentiable at every $x$ not belonging to $\Bbb Z$ and not differentiable at any $x$ belonging to $\Bbb Z$.

    Could I use a piecewise function to show this?

    • hhsaffar
      hhsaffar almost 10 years
      Yes, you can! What function you are thinking about?
    • hhsaffar
      hhsaffar almost 10 years
      Also $\lfloor{x}\rfloor$ can help you.
    • Emily
      Emily almost 10 years
      $\lfloor x \rfloor$ is neither continuous nor strictly increasing.
    • hhsaffar
      hhsaffar almost 10 years
      @Arkamis yes, but you can use it in combination of some other function.
    • user108626
      user108626 almost 10 years
      Thanks everyone for all your help! I had these sort of ideas but was not sure how to represent them!
    • hhsaffar
      hhsaffar almost 10 years
      @Arkamis I added a solution using $\lfloor{x}\rfloor$
  • Git Gud
    Git Gud almost 10 years
    Why make it CW?
  • André Nicolas
    André Nicolas almost 10 years
    I have no idea how it happened, I did not intend to make it CW.
  • André Nicolas
    André Nicolas almost 10 years
    Nice compact geometric description of the idea.
  • Git Gud
    Git Gud almost 10 years
    Delete it. I'll reupvote the new answer.
  • André Nicolas
    André Nicolas almost 10 years
    I tried, the machine will not accept the "new" answer, calls it a duplicate. Not exactly a tragedy! Anyway, I like the Arkamis version better.
  • Git Gud
    Git Gud almost 10 years
    Ah! Well, you can try to change the text on this one and then give a new answer if you want to.
  • Emily
    Emily almost 10 years
    Such a compliment I do not deserve!
  • user108626
    user108626 almost 10 years
    Thank you very much! This is very helpful!
  • hhsaffar
    hhsaffar almost 10 years
    $|x-\lfloor x+\frac 12 \rfloor|$ is actually the distance between $x$ and $Round(x)$.(e.g. $Round(3.1)=3,Round(3.7)=4$)