Problem 37 - 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. $$f(x)=\left\\{\begin{array}{ll} 2 x / x & x \neq 0 \\ 3 & x=0 \end{array}\right.$$

Step-by-Step Solution

Answer

The function is not continuous at \(x = 0\).

Step 1: Definition of Continuity

A function is continuous at a point \(x = c\) if the following three conditions are met:1. \(f(c)\) is defined.2. \(\lim_{{x \to c}} f(x)\) exists.3. \(\lim_{{x \to c}} f(x) = f(c)\).

Step 2: Analyze the function expression

The given function is \[f(x) = \begin{cases} \frac{2x}{x}, & x eq 0 \ 3, & x = 0 \end{cases}\]which simplifies to:\[f(x) = \begin{cases} 2, & x eq 0 \ 3, & x = 0 \end{cases}\].

Step 3: Check continuity at x = 0

First, find \(f(0)\): \[f(0) = 3\] which is defined.Next, evaluate \(\lim_{{x \to 0}} f(x)\):Since for \(x eq 0\), \(f(x) = 2\), \[\lim_{{x \to 0}} f(x) = 2\].Finally check if \(\lim_{{x \to 0}} f(x) = f(0)\):\(2 eq 3\), therefore, the function is not continuous at \(x = 0\).

Step 4: Conclusion on Continuity

The function is continuous everywhere except at \(x = 0\). The condition \(\lim_{{x \to 0}} f(x) = f(0)\) is not satisfied, therefore, the function is not continuous at this point.

Key Concepts

Limits and Continuity

Understanding the concept of limits and continuity is crucial in analyzing how functions behave. A function is deemed continuous at a point if it meets three specific criteria:

• The function value at that point exists.
• The limit of the function as it approaches that point from both directions must exist.
• The limit and the function value must be equal at that point.

In the context of the given exercise, these criteria are applied to ascertain whether a function maintains continuity across a range of values. If any of these conditions fail, particularly the match between the limit and the function value, we encounter a discontinuity.
This happens at a specific point where the smooth operation of the function is disrupted. The exercise involves checking whether a piecewise function is continuous at each point, especially where different rules apply as the values approach the designated point from either side.

Piecewise Functions

Piecewise functions involve different expressions or 'pieces' for different intervals of the variable. This means that instead of having a single formula for all values of the variable, the function might switch between formulas based on the variable's range.
In our exercise, the piecewise function is defined as:

• For values other than zero, the function follows the rule: \(f(x) = \frac{2x}{x} = 2\).
• At zero, the function uses: \(f(0) = 3\).

Analyzing piecewise functions for continuity involves examining each segment separately. The main focus is on the points where the function's rules change. Often, these transitions are the sites where continuity might be disrupted. Therefore, verifying limits at these points is a necessary step to confirm if the piecewise function behaves smoothly across its entire domain.

Discontinuity at a Point

A discontinuity in a function is like a break or gap at a certain point on its graph. This happens when the conditions for continuity are not met. Specifically, at a point of discontinuity:

• The value of the function might not be defined.
• The limit approaching the point from either direction might not exist or might not be equal to the function's value at that point.

In our function, there's a discontinuity at \(x = 0\). While \(f(0)\) is defined as 3, the limit of \(f(x)\) as \(x\) approaches 0 is 2. This mismatch between the limit and the function value shows that the function "jumps" at \(x = 0\).
Understanding these discontinuities is key to analyzing a function's behavior and helps us predict its characteristics and potential problem areas in its domain.