#AP Calculus AB/BC: Continuity Over an Interval 🚀

Hey there, future calculus master! Let's make sure you're totally solid on continuity over an interval. This is a big topic, and we're going to break it down so it feels like a piece of cake. 🍰

This topic is super important because it connects limits and functions. Expect to see it in both multiple-choice and free-response questions.

# 📈 Continuity on an Interval

The official definition from the College Board is that a function is continuous on an interval if it's continuous at every single point within that interval. Think of it like a smooth, unbroken road—no potholes, no sudden cliffs. 🛣️

For example, $f(x) = ln(3x)$ is continuous on its domain, which is $(0, \infty)$.

# 🏁 Continuity for Piecewise Functions

Piecewise functions are where things get a bit more interesting! We need to check continuity for each piece and at the points where the function changes its definition. It's like making sure each road segment connects perfectly to the next.

#⛳ Checking Domain Restrictions

Watch out for these common domain restrictions:

• Square Roots: $\sqrt{z}$ requires $z \geq 0$. No imaginary numbers allowed! 🚫

  For example, $f(x) = \sqrt{3x+1}$ has a domain of $[-\frac{1}{3}, \infty)$.

  

  The graph of $f(x) = \sqrt{3x+1}$ showing its domain.

• Rational Functions: $...