Can we differentiate any function anywhere?
Differentiation can only be applied to functions that 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 no 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 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 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 at x = 0.
These are the only kinds of non-differentiable behavior you will encounter, and you probably will not encounter many of these.
Now you have seen just about everything there is to say about differentiating functions of one variable. We next want to study how to apply this, and how to undo differentiation.