Continuity and Differentiability
Understanding the relationship between smooth curves and their rates of change.
Differentiability
While continuity means a graph can be drawn without lifting your pen, differentiability is a stricter condition. It means the graph is "smooth" at a point, without any sharp corners, cusps, or vertical tangents. A function is differentiable at a point if its derivative exists at that point.
Essentially, if you can zoom in on a point on the graph enough that it starts to look like a straight line, the function is differentiable there. The slope of this line is the derivative.
Key Idea: All differentiable functions are continuous, but not all continuous functions are differentiable.
The function f(x) = |x| is continuous everywhere, but it is not differentiable at x=0 because of the sharp corner.