Unit 1 Master Guide: Continuity in Calculus

Defining Continuity at a Point

In standard geometry, we often describe a continuous curve as one you can draw without lifting your pencil from the paper. However, in AP Calculus, we need a rigorous mathematical definition to prove continuity analytically.

For a function $f(x)$ to be continuous at a specific point $x = c$, three strict conditions must be met simultaneously. This is often referred to as the Definition of Continuity Checklist.

The Three-Part Definition

1. $f(c)$ must be defined: The value of the function at $x = c$ must exist (no holes or vertical asymptotes at the value itself).
2. $\lim{x \to c} f(x)$ must exist: The left-hand limit and the right-hand limit must approach the same finite value.
   \lim{x \to c^{-}} f(x) = \lim_{x o c^{+}} f(x)
3. The limit must equal the function value: The approached value (limit) must match the actual coordinate.
   \lim_{x \to c} f(x) = f(c)

If any of these three conditions fails, the function is discontinuous at $x = c$.

Types of Discontinuities

When continuity fails, it falls into specific categories. Understanding these classifications is essential for analyzing function behavior.

1. Removable Discontinuities (Holes)

These occur when the limit exists ($Condition #2$ is met), but the limit does not equal the function value ($Condition #3$ fails).

• Visual: A curve with a small open circle (hole).
• Cause: Usually caused by a common factor in the numerator and denominator of a rational function.
• Limit: $\lim_{x \to c} f(x)$ exists and is a real number.

2. Non-Removable Discontinuities

These are discontinuities that cannot be fixed by simply redefining a single point.

A. Jump Discontinuity

This happens when the left-hand limit and the right-hand limit both exist but are not equal.

• Visual: The graph "jumps" from one height to another at $x=c$.
• Cause: Common in piecewise functions or absolute value definitions like $y = \frac{|x|}{x}$.
• Limit: $\lim_{x \to c} f(x)$ does not exist (DNE) because the one-sided limits differ.

B. Infinite Discontinuity (Asymptotic)

This occurs when the function goes to positve or negative infinity as $x$ approaches $c$.

• Visual: A vertical asymptote.
• Cause: Division by zero where the numerator is non-zero.
• Limit: $\lim_{x \to c} f(x)$ does not exist (often described as approaching $\infty$ or $-\infty$).

C. Oscillating Discontinuity

This is a rare case where the function oscillates infinitely fast as it approaches a point (e.g., $f(x) = \sin(\frac{1}{x})$ at $x=0$). The limit does not exist purely due to oscillation.

Comparison Table

Confirming Continuity over an Interval

Once we understand continuity at a point, we expand the concept to an interval.

Open Intervals $(a, b)$

A function is continuous on an open interval $(a, b)$ if it is continuous at every single point inside that interval.

• Common Continuous Functions: Polynomials, exponential functions ($e^x$), sine, and cosine are continuous for all real numbers $(-\infty, \infty)$. Rational functions are continuous everywhere except where the denominator is zero.

Closed Intervals $[a, b]$

To claim continuity on a closed interval (including the endpoints), strictly checking the general limit isn't enough because limits generally require approaching from both sides. For endpoints, we use one-sided continuity.

A function is continuous on $[a, b]$ if:

1. It is continuous on the open interval $(a, b)$.
2. It is continuous from the right at $a$: $\lim_{x \to a^{+}} f(x) = f(a)$.
3. It is continuous from the left at $b$: $\lim_{x \to b^{-}} f(x) = f(b)$.

!A graph of a function on a closed interval [a, b] showing solid dots at the endpoints to indicate continuity from the right at a and left at b.

Removing Discontinuities

Only Removable Discontinuities can be "fixed." This process involves redefining the function or simplified algebraic manipulation to fill the "hole."

The Algebraic Approach

If a rational function yields the indeterminate form $\frac{0}{0}$ when evaluating a limit, there is likely a removable discontinuity (a hole) that corresponds to a common factor.

Worked Example:
Let $f(x) = \frac{x^2 - 9}{x - 3}$. Is there a discontinuity at $x=3$? Can we remove it?

1. Check $f(3)$: $\frac{3^2 - 9}{3 - 3} = \frac{0}{0}$. The function is undefined at $x=3$ (Condition 1 fails).

2. Find the Limit: Factor the numerator.
   \lim{x \to 3} \frac{(x-3)(x+3)}{x-3} Cancel the common factor $(x-3)$: \lim{x \to 3} (x+3) = 6
   Since the limit exists (it equals 6), this is a Removable Discontinuity.

3. Redefine the Function: To make the function continuous, we define a new function $g(x)$ that is the same as $f(x)$ everywhere except at the hole, where we plug the gap.

   g(x) = \begin{cases} \frac{x^2 - 9}{x - 3} & x \neq 3 \ 6 & x = 3 \end{cases}

This new function $g(x)$ is continuous at $x=3$ because now $g(3) = 6$ and $\lim_{x \to 3} g(x) = 6$.

Common Mistakes & Pitfalls

1. Confusing "Limit Exists" with "Continuous"

   • Mistake: Assuming that because the left and right limits match, the function is continuous.
   • Correction: A limit can exist even if there is a hole. You MUST check that the function value $f(c)$ also exists and equals that limit.

2. Assuming $\frac{0}{0}$ is an Asymptote

   • Mistake: Seeing a zero in the denominator and immediately claiming "Vertical Asymptote."
   • Correction: If you get $\frac{nonzero}{0}$, it is an asymptote (Infinite Discontinuity). If you get $\frac{0}{0}$, it is likely a hole (Removable Discontinuity). Always factor first.

3. Ignoring Endpoints on Intervals

   • Mistake: Applying standard two-sided limits to the endpoints of a closed interval.
   • Correction: At endpoints, calculate connection only from the interior of the interval (one-sided limits).