Introduction to Continuity

Welcome! In this chapter, we are moving from the "how-to" of limits to the "so what?" of limits. Continuity is one of the most important concepts in Calculus because most of the big theorems you will learn later—like the Intermediate Value Theorem or the Mean Value Theorem—only work if a function is continuous. Think of a continuous function as a path you can draw on paper without ever lifting your pencil. If you have to lift your pencil to jump over a gap, skip a hole, or follow an arrow to infinity, the function is discontinuous at that spot.

Defining Continuity at a Point

While the "pencil test" is great for your intuition, the AP exam requires a formal, three-step mathematical definition. For a function \(f(x)\) to be continuous at a specific point \(x = c\), it must pass all three of these tests: 1. The point must exist: \(f(c)\) is defined. (There isn't a hole or a vertical asymptote at exactly \(c\)). 2. The limit must exist: \(\lim_{x \to c} f(x)\) exists. (The left-hand and right-hand limits must meet at the same finite value). 3. The limit must equal the point: \(\lim_{x \to c} f(x) = f(c)\). (The "hole" is filled by the actual value of the function). Quick Tip: If any one of these three conditions fails, the function is discontinuous at \(x = c\). On Free-Response Questions (FRQs), you must explicitly check all three steps to justify continuity.

Key Takeaway: Continuity means "the value the function is approaching is exactly the same as the actual value at that spot."

Types of Discontinuities

When a function isn't continuous, it usually fails in one of three classic ways. The AP exam expects you to identify these by looking at a graph or an equation.

1. Removable Discontinuity (A "Hole")

A removable discontinuity occurs when the limit exists, but the function value either doesn't exist or is "somewhere else."
Visual: A straight line with a tiny open circle in the middle.
Algebraic Sign: Usually occurs when a factor cancels out in a rational function, such as \(f(x) = \frac{(x-2)(x+3)}{x-2}\). At \(x = 2\), there is a hole.

2. Jump Discontinuity

A jump discontinuity occurs when the left-hand limit and the right-hand limit are both finite but not equal to each other.
Visual: A "step" where the graph breaks and restarts at a different height.
Algebraic Sign: Frequently found in piecewise functions where the different "pieces" don't meet at the transition point.

3. Infinite Discontinuity

An infinite discontinuity occurs when the function approaches positive or negative infinity as \(x\) approaches \(c\).
Visual: A vertical asymptote.
Algebraic Sign: Occurs when the denominator of a simplified fraction is zero (and the numerator is not zero). For example, \(f(x) = \frac{1}{x}\) at \(x = 0\).

Did you know? We call it "removable" because you could "fix" the discontinuity just by redefining or filling in a single point. You can't "fix" a jump or an asymptote with just one dot!

Confirming Continuity over an Interval

A function is continuous on an interval if it is continuous at every single point within that interval.
Open Intervals \((a, b)\)
The function just needs to meet the 3-step criteria for every point between \(a\) and \(b\).
Closed Intervals \([a, b]\)
This is a bit more specific. For a function to be continuous on \([a, b]\), it must: 1. Be continuous on the open interval \((a, b)\). 2. Be continuous from the right at \(x = a\). This means \(\lim_{x \to a^+} f(x) = f(a)\). 3. Be continuous from the left at \(x = b\). This means \(\lim_{x \to b^-} f(x) = f(b)\). Analogy: Imagine a bridge. To be "safe" on the whole bridge, the road must be smooth in the middle, and the entrances at both ends must connect perfectly to the land.

Removing Discontinuities

As we mentioned, only removable discontinuities (holes) can be "fixed." To "remove" a discontinuity at \(x = c\), we create an extended function (usually a piecewise function) that defines the value at the hole to be equal to the limit. Step-by-Step Process: 1. Factor the numerator and denominator of the function. 2. Cancel the common factors. The value of \(x\) that made those factors zero is where the hole is. 3. Find the limit by plugging that \(x\) value into the remaining "simplified" version of the function. 4. Redefine the function so that \(f(c)\) equals that limit. Example: If \(f(x) = \frac{x^2 - 9}{x - 3}\), there is a hole at \(x = 3\).
Factoring gives \(\frac{(x-3)(x+3)}{x-3}\), which simplifies to \(x+3\).
The limit as \(x \to 3\) is \(3 + 3 = 6\).
To "remove" the discontinuity, we define a new function where \(f(3) = 6\).

Key Takeaway: Removing a discontinuity is just the process of "plugging the hole" with the limit value.

Common Mistakes to Avoid

  • Forgetting Step 3: Many students check that the limit exists and the point exists but forget to state that they are equal. You must write \(\lim_{x \to c} f(x) = f(c)\) to get full credit on the AP exam.
  • Confusing Jumps and Holes: If the denominator cancels out, it's a hole. If the denominator stays zero (and the top is non-zero), it's a vertical asymptote. Jumps usually only happen in piecewise or absolute value functions.
  • Calculator Over-reliance: Sometimes a calculator won't show a tiny hole in a graph. Always use algebraic manipulation to confirm what you see visually.

Note: This chapter focuses on identifying and defining continuity. To see how continuity applies to finding unknown constants in piecewise functions, refer to the chapter on "Determining limits algebraically". To see how continuity is used as a requirement for theorems, check the chapter on the "Intermediate Value Theorem".