Differentiability determines if something can be differentiated.
figure: An illustration of things that make a point not differentiable.
The black lines are your tangent lines, the things that you try to calculate when you differentiate. What makes a point not differentiable?
- Not continuous - if a point is not continuous, then it is impossible to calculate the slope of the point.
- Sharp turn - if the slope is always positive on one side and always negative on one side. So the slope is undefined.
- Infinite slope - if a point has infinite slope, the slope is undefined, and so the slope is also undefined.