Problem 36 - 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. $$f(x)=\left\\{\begin{array}{ll} x / x & x \neq 0 \\ 1 & x=0 \end{array}\right.$$

Step-by-Step Solution

Answer

The function is continuous for all \( x \).

Step 1: Review Function Definition

The function is defined as \( f(x) = \frac{x}{x} \) for \( x eq 0 \) and \( f(x) = 1 \) for \( x = 0 \). This can be simplified as \( f(x) = 1 \) for all \( x eq 0 \), since \( \frac{x}{x} = 1 \) when \( x eq 0 \). Thus, \( f(x) = 1 \) for all \( x \), but with an explicit definition at \( x = 0 \).

Step 2: Check Continuity Definition

A function \( f(x) \) is continuous at a point \( x = c \) if the following conditions hold: 1) \( f(c) \) is defined, 2) \( \lim_{x \to c} f(x) \) exists, and 3) \( \lim_{x \to c} f(x) = f(c) \). We need to check these conditions at \( x = 0 \), the only point where definition changes.

Step 3: Evaluate Limit as x Approaches 0

For \( x eq 0 \), \( f(x) = 1 \). Thus, the limit as \( x \to 0 \) is \( \lim_{x \to 0} f(x) = 1 \).

Step 4: Verify the Function Value at x = 0

The function value directly given at \( x = 0 \) is \( f(0) = 1 \).

Step 5: Compare Limit and Function Value

Since \( \lim_{x \to 0} f(x) = 1 \) and \( f(0) = 1 \), the function is continuous at \( x = 0 \) because the limit and the function value agree.

Step 6: Assess Continuity for Other Values of x

For all other \( x eq 0 \), \( f(x) \) is constantly \( 1 \), meaning it is continuous everywhere else by nature of a constant function.

Key Concepts

Piecewise Functions

A piecewise function is an interesting type of mathematical function. It is defined by different expressions based on different intervals of the independent variable. Imagine a cake cut into sections, each with its own unique flavor or icing! Each piece is delicious on its own, but together they make a complete cake.

For example, in the function provided in our exercise, we have:

• For all values of \(x\) not equal to zero, the function is defined as \(f(x) = \frac{x}{x} = 1\). This is because any number divided by itself (except zero) is 1.
• At the point \(x = 0\), the function is specifically defined as \(f(x) = 1\).

The idea here is that the function adjusts like a chameleon based on the value of \(x\), but it remains well-behaved and continuous, sticking to the value 1. The explicit definition at \(x = 0\) ensures the function is consistent across all inputs.

Limit of a Function

The limit of a function is an essential concept in calculus. It helps us understand the behavior of functions as they approach a particular point. You can think of trying to understand someone's mood by observing their expressions as they come closer to you!

In mathematical terms, the limit tells us what value a function approaches as the input gets very close to a specific number. For our piecewise function example:

• The limit as \(x\) approaches zero is calculated by observing the value that \(f(x)\) approaches as \(x\) gets closer and closer to zero, from both directions (left and right).
• Since \(f(x) = 1\) for all \(x eq 0\), the limit is quite simple: \(\lim_{x \to 0} f(x) = 1\).

This consistent behavior of the function around \(x = 0\) guarantees that the limit exists, and supports our journey to verify continuity.

Definition of Continuity

Continuity in mathematics describes how smooth a function graph is. If you can trace the graph of a function with a pen without lifting it, the function is continuous. It's like walking on a straight, seamless path without any breaks or jumps!

Mathematically, a function \(f(x)\) is considered continuous at a point \(x = c\) if:

• \(f(c)\) is defined.
• The limit \(\lim_{x \to c} f(x)\) exists.
• The limit at that point equals the function's value, meaning \(\lim_{x \to c} f(x) = f(c)\).

In our function, let's inspect the continuity at \(x = 0\):

• \(f(0) = 1\) is defined.
• The limit \(\lim_{x \to 0} f(x) = 1\) exists.
• Both the limit and the function value at \(x = 0\) match: \(\lim_{x \to 0} f(x) = f(0)\).

Hence, the function is continuous at \(x = 0\). For all other \(x eq 0\), the function is always 1, maintaining continuity. Thus, our piecewise function, though defined differently over intervals, manages to keep a smooth progression throughout.