second derivative does not exist at specified points

Question

Would any one give me an example or hint how to construct a function whose second derivative does not exist at some specified points say at n number of points. for first derivative I have the modulas function( i.e $|x|$), I need an example for second.

Answer 1

Really $|x|$ is just a convenient shorthand for what's really a piecewise defined function. If you allow piecewise defined functions, then it's rather easy to come up with examples.

$$f(x)=x^2,x>0$$ $$f(x)=-x^2,x<0$$

does the trick quite nicely at $x=0$. To make a function that has no second derivative at finitely/countably(?) many arbitrary points you can simply shift and add multiple functions like these.

Answer 2

Hint: integrate $|x|$.$\ \\\\\\\\\\\\$

Answer 3

$|x|$ also has a second derivative not defined at the origin. If you want something whose first derivative exists but second doesn't use integration (as suggested by others).

If you want to try something really fun think about going a step further the derivative of the absolute value function is $f(x)=\begin{cases} \phantom{-}1 & \text{if } x>0\\ -1 & \text{if } x<0 \end{cases}$

Integrate $\int f(x) dx$ to get $|x|$

In fact any step function integrated will give you a function that is continuous whose jump discontinuities become places the derivative of the integral is undefined.

For example take $\lfloor x \rfloor$ to be the greatest integer function. Then $f(x)=\int_0^x \lfloor t \rfloor dt$ is continuous everywhere is not differentiable at any integer value.