Question

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\left\\{\begin{array}{ll}{2 x-4,} & {x \neq 3} \\ {1,} & {x=3}\end{array}\right. $$

Short answer

The function \(f(x)\) is continuous on \((-∞, 3) and (3, +∞)\).

Step-by-step solution

Examine the domains of the individual functions

The function \(f(x)\) is composed of two functions: \(2x - 4\) which is valid when \(x\) does not equal \(3\), and \(1\) which is valid when \(x = 3\). So the domain of the entire function \(f(x)\) is \((-∞, 3) ∪ [3, +∞)\), which is all real numbers.

Check continuity of individual functions

Both \(2x - 4\) and \(1\) are basic functions (a linear function and a constant function) and are continuous over their respective domains. The function \(2x - 4\) is continuous on \((-∞, 3) and (3, +∞)\) and the function \(1\) is continuous at \(x = 3\).

Check the limit at \(x = 3\) from both sides

Calculate the limit of the function as \(x\) approaches \(3\) from the left: \n\[\lim_{{x}→{3^-}}f(x) = \lim_{{x}→{3^-}}2x - 4 = 2*3 - 4 = 2\]\nAnd calculate the limit of the function as \(x\) approaches \(3\) from the right: \n\[\lim_{{x}→{3^+}}f(x) = \lim_{{x}→{3^+}} 2x - 4 = 2*3 - 4 = 2\]\nSince the limits from both sides are not equal to the function value at \(x = 3\) which is \(1\), the function is not continuous at \(x = 3\).

State the intervals of continuity

Since the function is not continuous at \(x = 3\), the intervals on which the function is continuous are \((-∞, 3) and (3, +∞)\)

Key concepts

Piecewise Functions

A piecewise function is made up of different expressions based on the input variable's value. In the function given, \( f(x) = \begin{cases} 2x - 4, & x eq 3 \ 1, & x = 3 \end{cases} \), there are two rules: one for when \( x \) is not equal to three, and another for when \( x \) is exactly three. The way this function is structured allows it to behave differently at \( x = 3 \) compared to other \( x \)-values.
This method of defining functions is very useful when different conditions dictate different outcomes. In this case, the piecewise nature helps dictate the behavior at the single point \( x = 3 \). Such functions can often capture real-world scenarios where input values affect outputs in varied manners.

Limits

In calculus, limits help us understand what a function approaches as the input gets closer to a certain point. For the piecewise function in the exercise, we need to check the values of the function as \( x \) approaches \( 3 \) from both sides. This is crucial as it helps in determining the function's behavior and continuity at that particular point.
To find these limits:

• As \( x \to 3^- \) (approaching 3 from the left), the function behaves like \( 2x - 4 \). Hence, \( \lim_{{x} \to {3^-}} f(x) = 2 \).
• As \( x \to 3^+ \) (approaching 3 from the right), the function also follows \( 2x - 4 \). Therefore, \( \lim_{{x} \to {3^+}} f(x) = 2 \).

Even though both side limits are equal, they do not match the function value at \( x = 3 \), which is \( 1 \). This understanding of limits is pivotal in dealing with questions of continuity.

Domains of Functions

The domain of a function refers to all the possible input values (\( x \)-values) that will not cause the function to "break." For the function in question, \( f(x) \) is defined for all real numbers, as indicated by \((-∞, 3) \cup [3, +∞)\). This means that you can plug any real number into the function, and you will get a valid output.
However, because of the piecewise nature of the function, you need to look at where each piece applies. The linear piece \( 2x - 4 \) is valid everywhere except at \( x = 3 \), while the constant piece \( 1 \) specifically takes over at \( x = 3 \). Understanding the function's domain helps clarify where the function is behaving according to which piece, and it is essential for analyzing continuity.

Continuous Functions

A function is continuous at a point if: 1) it is defined at that point, 2) the limit of the function as \( x \to \text{point} \) exists, and 3) the limit equals the function's value at that point. For the function \( f(x) \), we check these at \( x = 3 \).
We know:

• The function value at \( x = 3 \) is \( 1 \).
• The limit as \( x \to 3^\pm \) is \( 2 \), as calculated earlier.

Since the limit approaching \( x = 3 \) from both sides doesn't equal the function value \( f(3) = 1 \), the function is not continuous at \( x = 3 \). It remains continuous at all other points within its domain. Therefore, the intervals of continuity are \((-∞, 3)\) and \((3, +∞)\). Understanding these principles is essential to handling the continuity of piecewise functions.