Problem 5 - Calculus and its applications

Question

Classify each statement as either true or false. If \(g\) is discontinuous at \(x=3,\) then \(g(3)\) must not exist.

Step-by-Step Solution

Answer

False; a function's value can exist at a point of discontinuity.

Step 1: Understand Discontinuity

A function is said to be discontinuous at a point if it is not continuous at that point. To understand what discontinuity means, it's important to know the conditions for continuity: a function is continuous at a point if the limit of the function as it approaches the point from both the left and right exists, is equal, and is equal to the function's value at that point.

Step 2: Evaluate Function Existence

The statement suggests that if a function is discontinuous at a point, the function's value at that point does not exist. However, discontinuity simply means that at least one of the three conditions for continuity is not met, which can include that the limit doesn't exist or isn't equal to the function's value, but it does not automatically imply the function's value at that point must not exist.

Step 3: Consider Counterexamples

Consider the piecewise function \( g(x) = \begin{cases} x^2 & \text{if } x eq 3 \ 5 & \text{if } x = 3 \end{cases} \). At \( x=3 \), \( g(3) = 5 \) clearly exists, yet the function is discontinuous because the limit of \( g(x) \) as \( x \) approaches 3 is 9, not 5. This shows a function can be discontinuous even if the function's value at the point exists.

Key Concepts

Understanding Discontinuity

A function experiencing discontinuity at a point simply means it is not smooth or unbroken at that point. For a function to be continuous at a particular point, three key conditions should be satisfied:

• The function must be defined at the point, meaning the value at that point, like \( g(3) \) in our example, should exist.
• The limit of the function as it approaches the point from the left must exist.
• The limit of the function as it approaches from the right should match the left-side limit and should also match the function's value at the point.

If any of these conditions are not met, the function is considered discontinuous at that point. However, be careful, the existence of the function's value at a point does not automatically imply continuity. It is just one of the necessary conditions.

Understanding Piecewise Functions

Piecewise functions are quite like their name—they are constructed from "pieces" of different functions. These functions use various expressions based on the input's interval. The function switches from one expression to another depending on which segment the input falls into.

An example of a piecewise function is:\( g(x) = \begin{cases} x^2 & \text{if } x eq 3 \ 5 & \text{if } x = 3 \end{cases}\)
Here, the function behaves differently when \( x \) is exactly 3 as compared to other values. For \( x = 3 \), the value of \( g(x) \) is 5. Outside of this point, it follows the rule \( x^2 \). This highlights how each "piece" of the function can have its distinct expression, yet all combine to form the whole function.
Piecewise functions are particularly useful in modeling real-world phenomena where a single formula can't accurately describe the entire scenario. The key to understanding these functions is to examine each piece, along with any boundary transitions, carefully.

Limits and Continuity

A key aspect of a function's behavior is how it approaches a point; this is where limits come into play. The limit of a function refers to the value the function approaches as the input gets closer to a specific point. Limits help us understand potential values, even when a function doesn't actually hit that value at the point itself.

For a function \( f(x) \) to be continuous at a point \( c \), the following must be true:

• The limit as \( x \) approaches \( c \) from the left (\( \lim_{x \to c^-} f(x) \)) must exist.
• The limit as \( x \) approaches \( c \) from the right (\( \lim_{x \to c^+} f(x) \)) should also exist and match the left-hand limit.
• This common limit value should be equal to the function's value at \( c \), that is, \( f(c) \).

If any of these conditions fail, we find the function exhibits discontinuity at that point. In many cases, calculating these limits is essential for identifying the type and reason for any discontinuity. Limits offer a detailed view of the transition in a function's behavior, allowing us to uncover the complex nature of continuity.