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

1 Answer
Jul 23, 2016

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#