Problem 45 - Calculus

Question

Are the statements true or false? Give an explanation for your answer. If a function is not continuous at a point, then it is not defined at that point.

Step-by-Step Solution

Answer

False, a function can be defined but not continuous.

Step 1: Understanding Continuity

The function is continuous at a point if the function is defined at that point and the limit of the function as it approaches that point from both sides equals the function's value at that point.

Step 2: Analyzing the Statement

The statement is 'If a function is not continuous at a point, then it is not defined at that point.' This suggests that discontinuity implies the function is undefined at that point. A function can be not continuous for other reasons, such as the limit not existing, the left-hand and right-hand limits not matching, or the limit existing but not matching the function's value, while still being defined.

Step 3: Counterexample Analysis

Consider the function \[ f(x) = \begin{cases} x^2 & \text{if } x eq 0 \ 1 & \text{if } x = 0 \end{cases} \]Here, the function is defined at \( x = 0 \). However, it is not continuous at \( x = 0 \) because \( \lim_{x \to 0} f(x) = 0 \), which does not equal \( f(0) = 1 \). So, the function is defined but not continuous.

Step 4: Conclusion

The given statement is incorrect because a function can be defined at a point but still not continuous there, as shown by the counterexample.

Key Concepts

Function Definition

A function is like a special machine that turns input (usually numbers) into output according to a specific rule. This rule determines the relationship between inputs and outputs. For example, a simple function could be doubling a number, where if you input 3, the output would be 6. Functions are fundamental in mathematics because they help us describe and analyze relationships between different quantities.
To understand whether a function is well-defined, it's important to check the following:

• Every input should correspond to exactly one output.
• The function must cover all the inputs we are considering (usually within a specific set like real numbers).

Whether a function is defined at a particular point means you can "plug in" this input into the function's formula and get a sensible output. For instance, if we have a function formula that uses division, it won't be defined where the denominator is zero since division by zero is undefined.

Limits in Calculus

In calculus, limits help us analyze the behavior of functions as they approach certain points. Imagine you're walking closer and closer to the edge of a cliff; the cliff's edge represents the point you are approaching, and how you get there is analogous to a limit.
The limit of a function at a point is the value that the function approaches as its input gets closer to that point. More formally, if we say that the limit of a function \(f(x)\) as \(x\) approaches \(a\) is \(L\), written as \(\lim_{x \to a} f(x) = L\), it means that by getting sufficiently close to \(a\), \(f(x)\) gets close to \(L\).
Here's why limits are important:

• They help us understand the behavior of functions near points where they might not be easy to calculate directly, like at discontinuities.
• They are foundational for defining derivatives and integrals, which are central in calculus.

Knowing limits let us determine function continuity, and they play a crucial role in assessing points of discontinuity.

Discontinuity Points

A discontinuity point in a function occurs where the function isn't smooth or "breaks" in some way. There are several scenarios where this might happen:

• The function might "jump," as seen in step functions or piecewise functions.
• The left-hand limit (as it approaches from the left) and right-hand limit (as it approaches from the right) might not agree.
• The limit might not exist at all due to chaotic oscillation or infinite behavior.
• The limit exists, but it does not equal the function's value at that point.

In the exercise's example, the function \( f(x) = \begin{cases} x^2 & \text{if } x eq 0 \ 1 & \text{if } x = 0 \end{cases} \) shows discontinuity at \(x = 0\) because the limit as \(x\) approaches zero is different from the value of the function at zero, even though the point \(x = 0\) is defined.
Understanding these discontinuity points is crucial in calculus as they mark changes in function behavior, affecting the results of integration and differentiation.