Welcome back to Professor Baker's Math Class! In this session, we are diving deep into Chapter 2-5, which focuses on the concept of Continuity. This is a bridge topic that connects limits to the more advanced derivative work we will do later. If you have ever wondered how to prove a graph doesn't have any holes or jumps, this is the chapter for you.

1. The Definition of Continuity

At its core, a function $f$ is continuous at a number $a$ if three conditions meet in a perfect harmony. As seen in the first page of our notes, the definition is:

$$\lim_{x \to a} f(x) = f(a)$$

This single equation actually tells us three things:

  • The function is defined at $a$ ($f(a)$ exists).
  • The limit as $x$ approaches $a$ exists (the left and right sides match).
  • The limit equals the function value.

If any of these break, we have a discontinuity. In class, we looked at graphs to identify Removable Discontinuities (holes), Jump Discontinuities (breaks in the line), and Infinite Discontinuities (vertical asymptotes).

2. Repairing Removable Discontinuities

One of the most common algebraic problems you will face is finding a value to "plug the hole" in a function. We often see rational functions that look like they are undefined at a certain point, such as:

$$f(x) = \frac{5x^3 - 20x - 25x}{x-5}$$

To make this function continuous at $x=5$, we cannot simply plug in 5 because we get a division by zero. Instead, we use limits:

  1. Factor the numerator.
  2. Cancel the common factor (the term causing the zero in the denominator).
  3. Evaluate the limit of the simplified expression.

The result of that limit is exactly where we should define the function value to make it continuous!

3. Finding Parameters for Continuity

A major theme in this chapter is working with piecewise functions. You might be asked to find a constant $k$ that makes a function continuous everywhere. The strategy here is straightforward:

Set the Left Limit equal to the Right Limit.

For example, if a function changes behavior at $x=3$, you would set the equation for $x < 3$ equal to the equation for $x > 3$, plug in $x=3$, and solve for $k$. This ensures the two pieces of the graph meet at the exact same point, preventing a jump discontinuity.

4. The Intermediate Value Theorem (IVT)

The IVT is a powerful theoretical tool. It states that if a function $f$ is continuous on a closed interval $[a, b]$, and $N$ is a number between $f(a)$ and $f(b)$, then there exists at least one number $c$ in $(a, b)$ such that $f(c) = N$.

In plain English: If a continuous function goes from a negative value to a positive value (or vice versa), it must cross zero at some point. You cannot skip values if the line doesn't break!

Note: Always check if the function is continuous on the specific interval first. If the function has a vertical asymptote (like $\frac{5}{x-4}$ at $x=4$) inside the interval, the IVT cannot be applied.

5. Trigonometric Limits and Continuity

Finally, we wrapped up with trigonometric functions. The beauty of continuity is that it makes evaluating limits much easier. Since functions like $\sin(x)$ and $\cos(x)$ are continuous everywhere, you can often use direct substitution.

For example:

$$\lim_{t \to -\frac{\pi}{2}} \frac{\cos(2t)}{3t + \sin(\cos(t))}$$

As long as the denominator isn't zero, you can simply plug in $t = -\frac{\pi}{2}$ to solve. Don't let the complex appearance of trig functions scare you—trust the continuity!

Keep practicing those piecewise algebra problems, and I'll see you in the next class!