Confirming Continuity over an Interval - AP Calculus BC Study Guide

The Core Idea: Confirming Continuity over an Interval

This topic expands the concept of continuity from a single point to an entire interval on the x-axis. The fundamental goal is to determine if a function's graph can be drawn over a specified interval without lifting your pencil. To do this, we don't test every single point, which would be impossible. Instead, we identify all potential "breaks"—points of discontinuity—for the function. If none of these breaks fall within the given interval, we can confirm the function is continuous over that entire stretch.

The process relies on knowing that many standard function types (like polynomials, rational functions, and trigonometric functions) are inherently continuous everywhere they are defined. Therefore, the task of confirming continuity over an interval simplifies to first finding the function's domain and then checking if the specified interval is a subset of that domain.

Key Definitions

Based on the Essential Knowledge, the core principles for confirming continuity over an interval are as follows:

1. Continuity on an Interval

A function $f(x)$ is defined as continuous on an interval if and only if it is continuous at every point within that interval. For a closed interval $[a,b]$, this includes continuity from the right at $x=a$ and continuity from the left at $x=b$.

2. Continuity of Standard Functions

Certain families of functions are continuous at all points in their domains. For these functions, any discontinuity can only occur at a value of $x$ where the function is undefined. The key function types are:

• Polynomial functions: Continuous for all real numbers, $(-\infty ,\infty )$.

• Rational functions: Continuous everywhere except at x-values that make the denominator zero.

• Power functions: (e.g., $x^n$, $\sqrt[n]{x}$) Continuous on their natural domains. Pay close attention to even roots, which require a non-negative radicand.

• Exponential functions: (e.g., $a^x,e^x$) Continuous for all real numbers, $(-\infty ,\infty )$.

• Logarithmic functions: (e.g., ${\log }_b(x),\ln (x)$) Continuous for all positive numbers (i.e., where the argument of the logarithm is greater than zero).

• Trigonometric functions:

  • $f(x)=\sin (x)$ and $f(x)=\cos (x)$ are continuous for all real numbers, $(-\infty ,\infty )$.

  • $f(x)=\tan (x),\sec (x),\csc (x),\cot (x)$ are continuous everywhere in their domains, with discontinuities at their vertical asymptotes.

Understanding the Domain

The central task in confirming continuity over an interval for the standard functions listed above is to first identify the function's domain. The question, "Is $f(x)$ continuous on the interval $(a,b)$?" becomes "Is the interval $(a,b)$ fully contained within the domain of $f(x)$?".

To do this, you must be proficient in finding the domains of functions, which typically involves looking for two main types of restrictions:

1. Division by Zero: For rational functions of the form $f(x)=\frac{p(x)}{q(x)}$, the domain consists of all real numbers except for the values of $x$ that make the denominator $q(x)=0$. These x-values correspond to points of discontinuity (either holes or vertical asymptotes).

2. Even Roots and Logarithms:

   • For functions involving an even root, such as $\sqrt{g(x)}$, the expression inside the root must be non-negative. The domain is restricted to x-values for which $g(x)\ge 0$.

   • For functions involving a logarithm, such as $\ln (g(x))$, the argument of the logarithm must be strictly positive. The domain is restricted to x-values for which $g(x)>0$.

Once you have determined all x-values where the function is discontinuous, you simply check if any of these values lie within the specific interval you are asked to consider.

Core Concepts & Rules

• A function is continuous across an interval if it is continuous at every individual point in that interval.

• To verify continuity on an interval, the primary strategy is to identify all points of discontinuity for the function.

• If none of the function's points of discontinuity are located within the specified interval, then the function is continuous on that interval.

• For polynomial, rational, power, exponential, logarithmic, and trigonometric functions, points of discontinuity only occur at x-values that are not in the function's domain.

• The process is therefore: 1) Find the domain of the function. 2) Compare the given interval to the domain.

Step-by-Step Example 1: Rational Function

Problem: Determine if the function $f(x)=\frac{x^2-9}{x^2-x-6}$ is continuous on the interval $[0,2]$. Is it continuous on the interval $[0,4]$?

Step 1: Identify Potential Discontinuities by Finding the Domain

The function $f(x)$ is a rational function. Discontinuities will occur where the denominator is equal to zero.

Set the denominator to zero and solve for $x$:

$$x^2-x-6=0$$

$$(x-3)(x+2)=0$$

$$x=3\text{or}x=-2$$

The function is discontinuous at $x=3$ and $x=-2$. The domain of $f(x)$ is $(-\infty ,-2)\cup (-2,3)\cup (3,\infty )$.

Step 2: Check the First Interval, $[0,2]$

Compare the points of discontinuity ($x=3,x=-2$) with the interval $[0,2]$.

• Is $x=3$ in the interval $[0,2]$? No.

• Is $x=-2$ in the interval $[0,2]$? No.

Since neither point of discontinuity lies within the interval $[0,2]$, the function is continuous on $[0,2]$.

Step 3: Check the Second Interval, $[0,4]$

Compare the points of discontinuity ($x=3,x=-2$) with the interval $[0,4]$.

• Is $x=3$ in the interval $[0,4]$? Yes.

• Is $x=-2$ in the interval $[0,4]$? No.

Since the point of discontinuity $x=3$ lies within the interval $[0,4]$, the function is not continuous on $[0,4]$.

Step-by-Step Example 2: Function with Logarithms and Roots

Problem: On which of the following intervals is the function $g(x)=\sqrt{x-1}+\ln (5-x)$ continuous?

A) $(1,5)$

B) $[1,5]$

C) $(-\infty ,1]$

D) $[5,\infty )$

Step 1: Determine the Domain of Each Part of the Function

The function $g(x)$ is composed of a square root term and a logarithmic term. For $g(x)$ to be defined, the conditions for both terms must be met simultaneously.

• For the square root term $\sqrt{x-1}$: The radicand must be non-negative.

$$x-1\ge 0⟹x\ge 1$$

• For the logarithmic term $\ln (5-x)$: The argument must be strictly positive.

$$5-x>0⟹5>x⟹x<5$$

Step 2: Combine the Domain Restrictions

We need to find the values of $x$ that satisfy both$x\ge 1$ and $x<5$.

Combining these gives the interval $[1,5)$.

The domain of $g(x)$ is $[1,5)$.

Step 3: Evaluate the Given Intervals

The function $g(x)$ is continuous on its entire domain, $[1,5)$. We now check which of the given answer choices is a subset of this domain.

• A) $(1,5)$: Is $(1,5)$ a subset of $[1,5)$? Yes. All numbers in $(1,5)$ are also in $[1,5)$.

• B) $[1,5]$: Is $[1,5]$ a subset of $[1,5)$? No, because the value $x=5$ is in $[1,5]$ but is not in the domain of $g(x)$ (it would cause $\ln (0)$).

• C) $(-\infty ,1]$: Is $(-\infty ,1]$ a subset of $[1,5)$? No.

• D) $[5,\infty )$: Is $[5,\infty )$ a subset of $[1,5)$? No.

Conclusion: The function $g(x)$ is continuous on the interval $(1,5)$. The correct choice is A.

Using Your Calculator

This topic is primarily analytical, meaning it must be solved by hand using algebraic reasoning about function domains. A calculator is not used to find the answer directly but can be an excellent tool for checking your work.

To verify your conclusion about an interval's continuity:

1. Graph the Function: Enter the function into your calculator's Y= editor.

2. Set the Window: Adjust the viewing window (WINDOW button) to match the interval in question. For Example 1 on the interval $[0,4]$, you might set $Xmin=0$, $Xmax=4$.

3. Visually Inspect: Graph the function and look for any breaks, holes, or vertical asymptotes within the window. For $f(x)=\frac{x^2-9}{x^2-x-6}$, you would see a vertical asymptote at $x=3$, confirming that the function is not continuous on $[0,4]$.

4. Use the Table: Go to TABLE (2nd + GRAPH). Scroll to the x-values where you identified discontinuities. You should see an ERROR in the $Y$ column, confirming that the function is undefined at that point.

AP Exam Quick Hit

Common Question Types

• Identifying a Continuous Interval: Given a function with domain restrictions (e.g., rational, logarithmic, or root function), you will be asked to select the interval from a list of options on which the function is continuous. (See Example 2 above).

  • Example: "On which of the following intervals is $f(x)=\frac{\ln (x)}{x-2}$ continuous? (A) $(0,\infty )$ (B) $(0,2)$ (C) $[0,2)$ (D) $(-\infty ,2)$$"

• Justifying Continuity for Piecewise Functions: You may be given a piecewise function and asked if it is continuous on an interval that contains the "seam" where the pieces meet. This requires checking the continuity of each piece on its sub-interval and checking the three-part definition of continuity at the seam.

  • Example: "Let $f(x)=\{\begin{array}{ll}{x}^2& x<1\\ \sqrt{x}& x\ge 1\end{array}$. Is $f(x)$ continuous on the interval $[0,5]$? Justify your answer."

Common Mistakes

• Ignoring the Interval: A very common error is to correctly find a point of discontinuity, for instance at $x=5$, but then incorrectly state the function is not continuous on an interval like $[0,4]$ that does not even contain the point of discontinuity. Always check if your discontinuity is inside the interval of interest.

• Domain Errors for Logarithms: Forgetting that the argument of a natural logarithm, $\ln (g(x))$, must be strictly positive ($g(x)>0$), not non-negative ($g(x)\ge 0$).

• Domain Errors for Square Roots: Confusing the rule for even roots with logarithms. The radicand of a square root, $\sqrt{g(x)}$, must be non-negative ($g(x)\ge 0$).

• Incomplete Factoring: When finding discontinuities in a rational function, students sometimes make algebraic errors when factoring the denominator, causing them to miss one or more points of discontinuity. For example, factoring $x^2-9$ as $(x-3)$ and forgetting the $(x+3)$ factor.