Problem 34 - Calculus

Question

In Problems \(34-37\), is the function continuous for all \(x ?\) If not, say where it is not continuous and explain in what way the definition of continuity is not satisfied. $$f(x)=1 / x$$

Step-by-Step Solution

Answer

The function is not continuous at \( x = 0 \) because it is undefined there, and the limit does not exist.

Step 1: Define Continuity

A function is continuous at a point if three conditions are met: 1) The function is defined at the point. 2) The limit of the function as it approaches the point exists. 3) The limit of the function as it approaches the point is equal to the function's value at that point.

Step 2: Identify Domain and Points of Discontinuity

The given function is \( f(x) = \frac{1}{x} \). It is undefined at \( x = 0 \) because division by zero is not possible. Therefore, the domain of \( f(x) \) is all real numbers except \( x = 0 \).

Step 3: Check Continuity at Point of Discontinuity

Continuity needs to be checked at \( x = 0 \), where the function might be discontinuous. Since \( f(x) \) is not defined at \( x = 0 \), condition 1 of the definition of continuity is not satisfied. Therefore, \( f(x) \) is not continuous at \( x = 0 \).

Step 4: Assess Limit at Discontinuity Point

As \( x \) approaches 0, the limit of \( f(x) = \frac{1}{x} \) does not exist because the function approaches infinity as \( x \to 0^+ \) and negative infinity as \( x \to 0^- \). Therefore, condition 2 of continuity is also not satisfied at \( x = 0 \).

Key Concepts

Discontinuous Functions

In mathematics, a function is said to be continuous at a point if it doesn't break or jump at that location. It is like drawing a line without lifting your pencil off the paper. However, some functions like \( f(x) = \frac{1}{x} \) have issues at certain points and are termed as discontinuous. Discontinuous functions are those for which the definition of continuity is not fully satisfied at some points in their domain. Let's revisit the conditions for continuity:- **Defined at the point:** The function must have a value at that specific point.- **Limit exists:** As you approach the point from both sides, the values should converge to a single value.- **Limit matches the function's value:** The value of the function at the point should be equal to this limit.For \( f(x) = \frac{1}{x} \), it becomes discontinuous at \( x = 0 \) because the function isn't defined there. You cannot divide by zero. Additionally, it fails the limit condition as it approaches infinity. This makes it a classic case of discontinuity.

Limits and Infinity

The concept of limits is central to calculus and helps in understanding the behavior of functions at particular points. When you say \( \lim_{{x\to a}}f(x) \), it means you're interested in what value \( f(x) \) is approaching as \( x \) gets closer to \( a \). For \( f(x) = \frac{1}{x} \), let's see what happens when \( x \) approaches 0:- **Approaching from the positive side (\( x \to 0^+ \)):** As \( x \) gets smaller but positive, \( f(x) \) grows larger. You will see it climbing towards infinity.- **Approaching from the negative side (\( x \to 0^- \)):** As \( x \) gets close to zero from the negative direction, \( f(x) \) results in increasingly negative values - shooting off towards negative infinity.Since the values of \( f(x) \) do not converge to a single number from both sides of 0, the limit does not exist. Both scenarios confirm that the limit at \( x = 0 \) doesn't exist, making the function not continuous at \( x = 0 \).

Domain and Range

Every function has a domain and a range, which tell us what values \( x \) can take and what values \( f(x) \) can produce, respectively. **Domain:**- For \( f(x) = \frac{1}{x} \), the main restriction comes from avoiding division by zero. Therefore, its domain is all real numbers except \( x = 0 \). You can say: - Domain: \( \{x \in \mathbb{R} \mid x eq 0\} \)**Range:**- As for the range, \( f(x) \) can take any real number except zero. That's because as \( x \) moves further from zero in either direction, \( f(x) \) grows larger in magnitude, covering all numbers but 0.- Range: \( \{y \in \mathbb{R} \mid y eq 0\} \)Knowing the domain and range helps anticipate discontinuities and sketch the graph, providing insights into the behavior of the function across different segments of its domain.