Question

What does it mean if a function is not differentiable at a point?

Answer 1

It means the function doesn't look like a line given a short enough interval.

Explanation:

Let's say we have a function `f(x)` and we're talking about the point `(a, f(a))`.

Now, the basic concept of derivation is that given a short enough interval, the function will look like a line, and when it does we can "find the slope of that line", but with more mathematical rigor.

So, sometimes we have a function like `|x|`, which just changes orientations abruptly, making a harsh v-shape. That means the point of the v-shape, in this case `(0,0)`, the function doesn't look like a line at a short enough interval, it will always look like a smaller and smaller and smaller v-shape.

Sometimes we have a function that just has a discontinuity somewhere, like, for example the signum function, which tells you if a number is negative or positive, for `x < 0`, it's `-1`, for `x >= 0` it's `1`. As there's this leap out of nowhere, it won't look a line.

We can also have erratic functions like, 0 for rational numbers and 1 for irrational numbers, those are so crazy they'll never look lines.

Or, either the function or its derivative can simply be undefined at that point, for example, the functions `1/x` and `root()x`. `1/x` is not defined at `x = 0`, and the derivative of `root()x`, `x^(-1/2)` isn't either. So neither function can be derivated when `x= 0`