Welcome back to Professor Baker's Math Class! Today, we are diving into Section 2.5: Continuity. As always, make sure you "Press Record" mentally and physically, because this concept bridges the gap between limits and the derivatives we will cover soon.

What is Continuity?

Intuitively, a function is continuous if you can draw its graph without lifting your pencil from the paper. However, in Calculus, we need a precise mathematical definition. According to the class notes, a function $f$ is continuous at a number $a$ if:

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

This single equation actually implies a three-step checklist. For a function to be continuous at $a$, all three of the following must be true:

  1. $f(a)$ is defined: The point must exist in the domain.
  2. $\lim_{x \to a} f(x)$ exists: The left-hand limit and right-hand limit must approach the same number.
  3. The limit equals the function value: The value you approach must be the value you actually land on.

Types of Discontinuities

When a function fails one of these tests, it is discontinuous. As we saw in the lecture graphs, there are three main types:

  • Removable Discontinuity: Often a "hole" in the graph. The limit exists, but $f(a)$ is undefined or differs from the limit. (Example: rational functions that simplify).
  • Jump Discontinuity: The function breaks and jumps to a new height. The left and right limits exist but are not equal. (Example: Piecewise functions or step functions like $f(x) = [[x]]$).
  • Infinite Discontinuity: The function shoots off to positive or negative infinity (vertical asymptote).

Continuity on an Interval & Domains

We also look at continuity over an entire interval. Many of the functions you know and love are continuous at every number in their domains. These include:

  • Polynomials and Rational Functions
  • Root Functions
  • Trigonometric and Inverse Trig Functions
  • Exponential and Logarithmic Functions

This is great news for calculating limits! If you know a function is continuous at a point, you can evaluate the limit simply by direct substitution.

The Intermediate Value Theorem (IVT)

Finally, we cover a powerful tool called the Intermediate Value Theorem. It states that if $f$ is continuous on a closed interval $[a, b]$, and $N$ is any number strictly between $f(a)$ and $f(b)$, then there must exist a number $c$ in $(a, b)$ such that $f(c) = N$.

Why is this useful? It helps us prove that solutions to equations exist without actually solving them. For example, if we have the polynomial $4x^3 - 6x^2 + 3x - 2 = 0$:

  • We evaluate at $x=1$: $f(1) = -1$ (Negative)
  • We evaluate at $x=2$: $f(2) = 12$ (Positive)

Since the function is a continuous polynomial and changes from negative to positive between 1 and 2, the IVT guarantees there is a number $c$ between 1 and 2 where the function crosses zero.

Review the attached notes for the specific graphs and the example involving the domain of $\ln(x) + \tan^{-1}(x)$. Keep practicing those limits, and I'll see you in the next section!