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I started to learn little about the definition of derivative, and I come across the following function. What is the derivative of the following function $$ f(x) = \begin{cases} x-a,\ \ \text{if}\ x\geq a\\ x,\ \ \ \ \ \ \ \ \ \text{if}\ x<a \end{cases}. $$ The function is obviously not continuous (at $x=a$) and hence I believe that the derivative is $1$ for any $x\neq a$ and not defined at $x=a$. If I'm right, how can I modify the function such that it's derivative will be defined at $x=a$?

Thank you.

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You're correct that the derivative is not defined at $a$ (if $a\neq 0$). But you need to take a look again at what the derivative is when it is defined; are you sure that the derivative of the function $f(x)=x$ can be made to be any real number $a$ that you choose? –  Zev Chonoles Feb 25 '13 at 6:55
    
The derivative is not $a$; is that a typo? In order to be differentiable, first it must be continuous. What values of $a$ make the function continuous at $a$? You can take left-hand and right-hand limits to find out. –  Jonas Meyer Feb 25 '13 at 6:57
    
Yes, it is a typo (fixed). Thank you! –  user63875 Feb 25 '13 at 7:00

1 Answer 1

It is not possible to make this function differentiable at $a$ by modifying it only at $a$. Because the left- and right- limits do not agree, the function will not be continuous at $a$, no matter what you assign as its value at $a$. And not being continuous, it can't be differentiable.

But you can make the function differentiable everywhere by modifying it in a small neighborhood of $a$ - as small as you wish. In a picture, it looks like this: from

bad

you get

good

The method I used here is called mollification. To be precise, the second function is defined as

$$g(x)=\int_{-0.1}^{0.1} f(x-t) \,(10-100|t|)\,dt$$

which isn't the best mollification possible but is simple to write, and good enough to make the function once differentiable.

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+1 for the graph. –  Shuhao Cao Jul 16 '13 at 4:36

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