What does it mean if a function is not differentiable at a point?
1 Answer
It means the function doesn't look like a line given a short enough interval.
Explanation:
Let's say we have a function
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
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
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