Continuity and the Derivative
Understanding the deep connection between continuous functions and differentiable functions
Part 1: Continuity — A Quick Review
Before we explore the derivative, we need a solid understanding of continuity. Intuitively, a function is continuous if you can draw its graph without lifting your pencil.
Formal Definition of Continuity
A function $ f(x) $ is continuous at a point $ x = a $ if and only if all three conditions hold:
- $ f(a) $ is defined (the function has a value at $ a $)
- $ \displaystyle\lim_{x \to a} f(x) $ exists (the limit exists)
- $ \displaystyle\lim_{x \to a} f(x) = f(a) $ (the limit equals the function value)
If any of these conditions fails, the function has a discontinuity at $ x = a $.
Types of Discontinuities
Removable
A "hole" in the graph. The limit exists but doesn't equal $ f(a) $ (or $ f(a) $ is undefined).
Example: $ f(x) = \frac{x^2 - 1}{x - 1} $ at $ x = 1 $
Jump
Left and right limits exist but are not equal.
Example: Piecewise functions with a "step"
Infinite
The function approaches $ \pm\infty $ (vertical asymptote).
Example: $ f(x) = \frac{1}{x} $ at $ x = 0 $