What Does It Mean for a Function to Be Continuous?

A function is continuous if you can draw its graph without lifting your pencil! More formally, a function \( f(x) \) is continuous at a point \( a \) if:

1. \( f(a) \) is defined.
2. \( \lim_{x \to a} f(x) \) exists.
3. \( \lim_{x \to a} f(x) = f(a) \).

Types of Discontinuity

• Jump Discontinuity: The function "jumps" from one value to another.
• Removable Discontinuity: There's a hole in the graph.
• Infinite Discontinuity: The function goes to infinity at a point.

Why Is Continuity Important?

Continuity ensures that limits and derivatives exist and behave nicely. In real life, continuous functions describe smooth changes, like temperature over time.

How to Check Continuity

• Plug in the value.
• Find the left and right limits.
• See if everything matches up.