## 9.3 Non-Differentiable Functions

**Can we differentiate any function anywhere?**

Differentiation can only be applied to functions whose graphs look like straight lines in the vicinity of the
point at which you want to differentiate. After all, differentiating is finding the slope of the line it looks
like (the tangent line to the function we are considering) No tangent line means no derivative.

Also when the tangent line is straight vertical the derivative would be infinite and that is not good either.

**How and when does non-differentiability happen [at argument \(x\)]?**

Here are some ways:

1. The function jumps at \(x\), (is not continuous) like what happens at a step on a flight of stairs.

2. The function's graph has a kink, like the letter V has. The absolute value function, which is \(x\) when \(x\) is positive and \(-x\) when \(x\) is negative has a kink at \(x = 0\).

3. The function is unbounded and goes to infinity. The functions \(\frac{1}{x}\) and \(x ^{-2}\) do this at \(x = 0\). Notice that at the particular argument \(x = 0\), you have to divide by \(0\) to form this function, and dividing by \(0\) is not an acceptable operation, as we noted somewhere.

4. The function is totally bizarre: consider a function that is \(1\) for irrational numbers and \(0\) for rational numbers. This is bizarre.

5. The function can't be defined at argument \(x\). When we are talking about real functions the square root cannot be defined for negative \(x\) arguments.

6. The function can be defined and finite but its derivative can be infinite. An example is \(x^{1/3}\) at \(x = 0\).

7. The function can be defined and nice, but it can wiggle so much as to have no derivative. Try to
differentiate \(\sin\left(\frac{1}{x}\right)\) at \(x = 0\). This kind of behavior is called **an
Essential Singularity at \(x = 0\).**

These are the only kinds of non-differentiable behavior you will encounter for functions you can describe by a formula, and you probably will not encounter many of these.

Now you have seen almost everything there is to say about differentiating functions of one variable. There is a little bit more; namely, what goes on when you want to find the derivative of functions defined using power series, or using the inverse operation to differentiating. We will get to them later.

We next want to study how to apply this, and then how to invert the operation of differentiation.