# If a function has a continuous derivative, yet is disjoint at a point, do we say that point has a derivative?

There have been MANY variations of this question answered elsewhere, but I could find no example of this particular question, and it's important:

Say you have a piecewise function defined by different functions on different domain intervals. At the point where the two intervals meet:

-> The function value is well defined

-> The derivative of that function is the same from the left and the right, therefore it is continuous

-> But the function itself is not continuous at that point - it has a break.

Do we say the derivative exists at that point?

For a concrete example:

y = 0 if x < 0

y = cos x if x >= 0

In this example, at the point x=0, the derivative of y = cos x is - sin x = -sin 0 = 0. And the derivative of y = 0 is always 0. So the derivative, y', is continuous across x = 0. It is the same slope from the right or the left side.

BUT, at x = 0, the graph itself is at 1, but jumps to 0 on the left - it is defined at 0 but discontinuous.

So when a function is discontinuous (but defined) at a point, yet the derivative IS continuous at that point (same from both directions), do we say the derivative at that point exists? Note in this case it is not a multiple valued function - the point at 0 is well defined - it equals 1.

This question can be generalized: if at a given point, the Nth derivative of a function is not continuous, do we consider the (N+i)th derivative at that point for all i to also be undefined?

Thanks so much for your thoughts.

• Welcome to MSE! <> In a word, no. You can check that the definition, a limit of difference quotients, does not exist at that point (and are welcome to post your own answer if you like). Oct 10, 2023 at 22:34
• If your definition of the derivative is $\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$, note that the numerator is $f(x+h)-f(x)$, not $f(x+h)-\lim_{t \to x}f(t)$. So continuity is necessary. Oct 10, 2023 at 23:08

• Markdown isn't quite LaTeX, so we use asterisks to do italics; for example, *differentiable* yields differentiable. (That term is probably preferable to "derivable.") Oct 10, 2023 at 23:06