#AP Calculus AB/BC: Differentiability and Continuity - The Ultimate Review 🚀

Hey there, future calculus conqueror! Let's make sure you're totally prepped for anything the AP exam throws your way. This guide is designed to be your go-to resource for understanding differentiability and continuity, especially when time is tight. Let’s dive in!

#🔗 2.4 Connecting Differentiability and Continuity

#↗️ Continuity and Differentiability

Most functions you'll see in AP Calc are both continuous and differentiable, but it's crucial to know when things go sideways. Remember, for a function to be differentiable at a point, it must be smooth and look like a straight line when you zoom in super close. This line doesn't have to be horizontal!

Check out the graph of $\cos(x)$ below. Notice how zooming in at $(0,1)$ makes it look like a straight line. That’s differentiability in action!


Zooming into graph of $\cos(x)$

Graph created with Desmos

For a function to be differentiable at a point 'a', the left-hand limit of the derivative must equal the right-hand limit of the derivative, and both must equal the value of the derivative at that point:

$$\lim_{x\to a^{-}} f'(x) = \lim_{x\to a^{+}} f'(x) = f'(a)$$

#🚧 When is a Function NOT Differentiable?

A function is NOT differentiable if it has any of the following:

• Discontinuities (jumps, holes, or vertical asymptotes)
• Sharp Turns (corners or cusps)
• Vertical Tangents

Let's break down each scenario:

#🕳 Discontinuous Graphs

#Unequal Derivatives

If a graph has a discontinuity, the derivatives from the left and right sides will not match. Take a look at $g(x) = \frac{1}{x+2}$.


Graph of $g(x) = \frac{1}{x+2}$

Graph created with Desmos

Notice that $\lim_{x\to -2^{-}} g'(x) < 0$ and $\lim_{x\to -2^{+}} g'(x) > 0$. Since the derivative values are ...