Question

[Calculus, Derivatives] Can a function be differentiable at certain points even if the entire function is not differentiable?

This would be where, in a function where a point is not differentiable ( such as at x = -2 in f(x) = |x+2| ), could the other points (at x = 1, 5, 500, anything but x = -2) be considered differentiable despite the function as a whole not being differentiable due to that non-differentiable point?

(I figured since there can be continuity at one point, but not throughout an entire function, that it may be the same here even though continuity and differentiability are two different things?? Regardless, clarification is really appreciated! Even if it may turn out to be useless to know for the course or anything in general, I'd like to know.)

Answer 1

A function is differentiable if you can find the derivative at every point in its domain. In the case of f(x) = |x+2|, the function wouldn't be considered differentiable unless you specified a certain sub-interval such as (5,9) that doesn't include x = -2. Without clarifying the interval, the entire function overall is not differentiable even if there's only one point at issue here (because again we look at the entire domain). Though to be fair, you could easily say "the function f(x) = |x+2| is differentiable everywhere but x = -2" and would be correct. So it just depends on your wording really.