When is a curve differentiable?
0:00 // What is the definition of differentiability?
0:29 // Is a curve differentiable where it’s discontinuous?
1:31 // Differentiability implies continuity
2:12 // Continuity doesn’t necessarily imply differentiability
4:06 // Differentiability at a particular point or on a particular interval
4:50 // Open and closed intervals for differentiability
5:37 // Summary
When we talk about differentiability, it’s important to know that a function can be differentiable in general, differentiable over a particular interval, or differentiable at a specific point. In order for the function to be differentiable in general, it has to be differentiable at every single point in its domain. If there’s just a single point where the function isn’t differentiable, then we can’t call the entire curve differentiable. In that case, we could only say that the function is differentiable on intervals or at points that don’t include the points of non-differentiability.
So how do we determine if a function is differentiable at any particular point? Well, a function is only differentiable if it’s continuous. So if there’s a discontinuity at a point, the function by definition isn’t differentiable at that point. This applies to point discontinuities, jump discontinuities, and infinite/asymptotic discontinuities.
But there are also points where the function will be continuous, but still not differentiable. Remember, differentiability at a point means the derivative can be found there. If there derivative can’t be found, or if it’s undefined, then the function isn’t differentiable there.
So, for example, if the function has an infinitely steep slope at a particular point, and therefore a vertical tangent line there, then the derivative at that point is undefined. That means we can’t find the derivative, which means the function is not differentiable there. In the same way, we can’t find the derivative of a function at a corner or cusp in the graph, because the slope isn’t defined there, since the slope to the left of the point is different than the slope to the right of the point. Therefore, a function isn’t differentiable at a corner, either.
Therefore, in order for a function to be differentiable, it needs to be continuous, and it also needs to be free of vertical slopes and corners. Barring those problems, a function will be differentiable everywhere in its domain.
Music by: Nicolai Heidlas
Song title: Wings