Give an example of a continuous strictly increasing function g:R>R which is...
Solution 1
Here is a way to construct such a function.
 Draw $f(x)=x^3+x^2+x$.
 At every integer $n$, draw $g(x)$ between $n$ and $n+1$ as the secant from $f(n)$ to $f(n+1)$.
 $g(x)$ 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:
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 $f(x)=2^n1+2^n x$.
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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user108626
Updated on February 20, 2020Comments

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 almost 10 yearsYes, you can! What function you are thinking about?

hhsaffar almost 10 yearsAlso $\lfloor{x}\rfloor$ can help you.

Emily almost 10 years$\lfloor x \rfloor$ is neither continuous nor strictly increasing.

hhsaffar almost 10 years@Arkamis yes, but you can use it in combination of some other function.

user108626 almost 10 yearsThanks everyone for all your help! I had these sort of ideas but was not sure how to represent them!

hhsaffar almost 10 years@Arkamis I added a solution using $\lfloor{x}\rfloor$


Git Gud almost 10 yearsWhy make it CW?

André Nicolas almost 10 yearsI have no idea how it happened, I did not intend to make it CW.

André Nicolas almost 10 yearsNice compact geometric description of the idea.

Git Gud almost 10 yearsDelete it. I'll reupvote the new answer.

André Nicolas almost 10 yearsI 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 almost 10 yearsAh! Well, you can try to change the text on this one and then give a new answer if you want to.

Emily almost 10 yearsSuch a compliment I do not deserve!

user108626 almost 10 yearsThank you very much! This is very helpful!

hhsaffar almost 10 years$x\lfloor x+\frac 12 \rfloor$ is actually the distance between $x$ and $Round(x)$.(e.g. $Round(3.1)=3,Round(3.7)=4$)