Continuity and Differentiability

TL;DR

Continuity means a function's graph has no breaks or jumps, allowing you to draw it without lifting your pen. Differentiability means you can find a unique tangent line at every point, implying a smooth curve without sharp corners. These concepts are fundamental for understanding how functions behave and change.

1. The Mental Model

Imagine a function as a path on a graph. If you can walk along the path without ever needing to jump or step over a gap, it's continuous. If the path is also perfectly smooth, with no sharp turns or corners, it's differentiable.

2. The Core Material

What is Continuity?

A function $f(x)$ is continuous at a point $x=a$ if three conditions are met:
1. $f(a)$ exists: The function is defined at that point.
2. $\lim_{x \to a} f(x)$ exists: As you approach $a$ from both the left and right sides, the function values approach the same number. This is called the limit.
3. $\lim_{x \to a} f(x) = f(a)$: The limit you found is exactly equal to the function's value at that point.

Think of it like this: the point exists, the function agrees on where it's going (from both sides), and where it's going is exactly where it lands.

If any of these conditions fail, the function is discontinuous at $a$. Types of discontinuity include:
* Removable discontinuity (hole): The limit exists, but it doesn't equal $f(a)$ or $f(a)$ isn't defined.
* Jump discontinuity: The left and right-hand limits are different.
* Infinite discontinuity (vertical asymptote): The function goes off to infinity.

What is Differentiability?

A function $f(x)$ is differentiable at a point $x=a$ if its derivative $f'(a)$ exists at that point. The derivative at a point represents the instantaneous rate of change or the slope of the tangent line to the curve at that point.

For the derivative to exist at $x=a$, the following must be true:
1. The function must be continuous at $x=a$. This is a critical prerequisite. If it's not continuous, you can't draw a single, well-defined tangent.
2. The left-hand derivative must equal the right-hand derivative. This means the slope of the tangent line approaching from the left must be the same as the slope approaching from the right. If they're different, you'd have a sharp corner (like at the tip of a V-shape graph), and you can't draw a unique tangent.

Important Relationship:
* **If a function is differentiab