Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ f(t)=|t| $$

Step-by-Step Solution

Answer

The function \( f(t) = |t| \) is continuous at \( t = 3 \).

Step 1: Define Continuity

A function \( f(t) \) is continuous at a point \( t = a \) if the following three conditions are met:1. \( f(a) \) is defined.2. \( \lim_{{t \to a}} f(t) \) exists.3. \( \lim_{{t \to a}} f(t) = f(a) \).In this exercise, we will evaluate whether these conditions are met for \( f(t) = |t| \) at \( t = 3 \).

Step 2: Is the Function Defined at t = 3?

The function \( f(t) = |t| \) is defined for all real numbers, including \( t = 3 \). Therefore, \( f(3) = |3| = 3 \). The first condition of continuity is met.

Step 3: Calculate the Limit as t Approaches 3

Next, we evaluate \( \lim_{{t \to 3}} |t| \). The absolute value function is continuous, so the limit can be calculated by substitution:\[ \lim_{{t \to 3}} |t| = |3| = 3. \]Therefore, the second condition is satisfied because the limit exists.

Step 4: Compare Limit and Functional Value

Since we have both \( f(3) = 3 \) and \( \lim_{{t \to 3}} f(t) = 3 \), they are equal. Therefore, the third condition is met as well.

Key Concepts

Absolute Value Function

The absolute value function is a mathematical function used to describe the magnitude or distance of a number from zero without regard to its sign. It is typically notated as \(|x|\), where \(x\) is any real number. This function is particularly unique in its definition:

• If \(x\) is any positive number or zero, then \(|x| = x\).
• If \(x\) is negative, then the absolute value \(|x| = -x\).

Therefore, the function \(f(t) = |t|\) outputs the same value if \(t\) is positive or zero, and it switches the sign of \(t\) when it is negative. The graph of the absolute value function has a distinctive 'V' shape, reflecting its effect of "folding" negative values to their positive counterparts.
Understanding this fundamental behavior helps explain why the function maintains certain properties, such as continuity, as we will see shortly.

Limits

The concept of a limit is fundamental in calculus and is essential to understanding the behavior of functions as they approach a particular point. A limit describes the value that a function approaches as the input gets arbitrarily close to a certain point. In mathematical notation, the expression \(\lim_{{x \to a}} f(x)\) signifies the value that \(f(x)\) approaches as \(x\) moves closer to \(a\).
For continuous functions, like the absolute value function \(f(t) = |t|\), finding the limit usually involves straightforward substitution.
For instance, to find \(\lim_{{t \to 3}} |t|\), we substitute \(3\) into the function, yielding \(3\), which matches the function's value at that point. This behavior signifies that as \(t\) approaches \(3\), the function approaches the same value as \(f(t)\) itself at \(t = 3\).
However, for functions with potential discontinuities or that are defined piecewise, calculating limits may involve checking the limits from the left and right separately, ensuring that they converge to the same value.

Conditions for Continuity

To determine if a function is continuous at a given point, we need to verify three specific conditions. These conditions provide a rigorous framework to conclude whether a function behaves smoothly without any breaks at the given point.

• First, the function must be defined at the point. This means \(f(a)\) needs to have a real number value.
• Second, the limit of the function as it approaches the point must exist, expressed as \(\lim_{{x \to a}} f(x)\).
• Third, the limit taken as the input approaches the point must equal the function's defined value at that point, thereby \(\lim_{{x \to a}} f(x) = f(a)\).

In the case of \(f(t) = |t|\) at \(t = 3\), all these conditions are perfectly met: the function is defined at \(t = 3\) with \(f(3) = 3\), the limit as \(t\) approaches 3 is also 3, and the limit matches the function's value at \(3\).
Therefore, this function is continuous at that point.
Understanding these conditions helps ensure clarity when analyzing more complex functions where discontinuities may exist.