91Ó°ÊÓ

In calculus, understanding when a function is continuous or discontinuous is crucial. A continuous function has no breaks, jumps, or holes at any point within its domain. This means you can draw the function without lifting your pencil. Mathematically, a function is continuous at a point if the limit of the function as it approaches the point is equal to the function's value at that point. If there's a sudden change in value, we have a discontinuity.
A discontinuous function has at least one point where the limit does not equal the function’s value. Discontinuous points can be categorized as:

• Jump discontinuities: These occur when there's a sudden jump in the value of the function.
• Infinite discontinuities: These occur when the function goes to infinity at that point.
• Removable discontinuities: These occur when a limit exists but does not equal the function's value.
  Understanding these distinctions is pivotal for solving problems involving continuity and discontinuity in functions.

A piecewise function is a function defined by multiple sub-functions, each applying to a specific interval of the domain. These functions are helpful in modeling situations where the behavior of a function changes at different intervals.
Consider the function defined in the exercise:
\[ f(x) = \begin{cases} 1 & \text{if } x = \frac{1}{n}, \text{ where } n \in \mathbb{N} \ 0 & \text{otherwise} \end{cases} \]
Here, the function is ‘piecewise’ because it behaves differently at the discontinuous points compared to other points. For the given specific points \( x = \frac{1}{n} \) where \( n \) is a natural number, the function value is 1, and for all other points, it is 0. This makes the function discontinuous at those specific points.
Piecewise functions allow us to construct functions with desired properties, such as discontinuities at specific points, which are crucial in solving calculus problems where controlled discontinuities are needed.

Solving calculus problems often involves identifying patterns and logical step-by-step reasoning. Let's break down the approach for constructing the given functions with specific discontinuities.
Step 1: Understanding Discontinuity
Identify that a function is discontinuous at a point if there's a sudden jump.
Step 2: Identifying Points of Discontinuity
For part (a), the points are \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \). For part (b), it includes \( 0 \) along with the same points.
Step 3: Constructing the Function for Part (a)
Define the function as:
\[ f(x) = \begin{cases} 1 & \text{if } x = \frac{1}{n}, \text{ where } n \in \mathbb{N} \ 0 & \text{otherwise} \end{cases} \]
This makes the function discontinuous at each \( \frac{1}{n} \) and continuous elsewhere.
Step 4: Constructing the Function for Part (b)
Define the function as:
\[ g(x) = \begin{cases} 1 & \text{if } x = \frac{1}{n}, \text{ where } n \in \mathbb{N} \ 2 & \text{if } x = 0 \ 0 & \text{otherwise} \end{cases} \]
This function is discontinuous at each \( \frac{1}{n} \) and at \(0\).
Step 5: Verification
Verify by checking continuity at points other than \( \frac{1}{n} \) and \( 0 \). For these other points, both functions stay at zero, indicating no jumps.
This logical approach and organization of steps are key skills in calculus problem-solving, ensuring you can tackle various function-related problems efficiently.