Question

Explain the distinction between a continuous and a discontinuous function. Draw a graph showing an example of each type of function.

Short answer

A continuous function is a function with no breaks, gaps, or holes throughout its domain, and can be drawn as a continuous line without lifting your pen from the paper. A discontinuous function has breaks, gaps, or holes in its domain, making it impossible to draw as a continuous line. A continuous example is f(x) = 2x + 1, where its graph is a straight line with a slope of 2 and a y-intercept of 1, extending infinitely in both directions. A discontinuous example is the piecewise function f(x) = {x for x<0, 2 for x=0, -x for x>0}. The graph consists of two lines: y=x for x<0 and y=-x for x>0, with a hole at x=0, represented by an open circle at the point (0,2).

Step-by-step solution

1. Define Continuous and Discontinuous Functions

A continuous function is a function that has no breaks, gaps, or holes throughout its domain. In other words, you can draw the entire graph of the function without lifting your pen from the paper. Mathematically, a function f is continuous at a point x=a if the limit as x approaches a of f(x) is equal to f(a). A discontinuous function, on the other hand, is a function that has breaks, gaps, or holes in its domain. These interruptions in the function's graph make it impossible to draw the entire graph without lifting your pen from the paper. Mathematically, a function f is discontinuous at a point x=a if the limit as x approaches a of f(x) does not exist or is not equal to f(a).

2. Example of a Continuous Function

A common example of a continuous function is a linear function. The function f(x) = 2x + 1 is continuous as it has no breaks, gaps, or holes throughout its domain (which is all real numbers). The graph of this function is a straight line, which can be drawn without lifting your pen from the paper.

3. Example of a Discontinuous Function

A common example of a discontinuous function is a piecewise-defined function. Let's consider the following piecewise function: f(x) = \begin{cases} x, & x < 0 \\ 2, & x = 0 \\ -x, & x > 0 \end{cases} For x<0, the function is continuous. At x=0, the function has a "hole" with a value of 2 (since f(0)=2). For x>0, the function is continuous. However, due to the hole at x=0, the function is discontinuous.

4. Draw Graphs for Both Types of Functions

For the continuous function, f(x) = 2x + 1, draw a straight line with a slope of 2 and a y-intercept of 1. The graph should extend infinitely in both directions. For the discontinuous function, f(x) is defined piecewise: draw the line y=x for x<0, and draw the line y=-x for x>0. At x=0, there is a hole, which can be represented by an open circle at the point (0,2). In conclusion, a continuous function such as f(x) = 2x + 1 has no breaks or gaps in its domain, and its graph can be drawn without lifting the pen from the paper. In contrast, a discontinuous function like the piecewise function f(x) has a hole in its graph at x=0 and can't be drawn as a continuous line.

Key concepts

Piecewise-Defined Functions

Piecewise-defined functions are fascinating because they are defined by different expressions for different parts of their domains. This means, depending on the input value, the function might behave differently.

Imagine driving a car on a road with different speed limits in various zones. In the same way, a piecewise function changes its behavior at specified points. Consider the function \( f(x) = \begin{cases} x, & x < 0 \ 2, & x = 0 \ -x, & x > 0 \end{cases} \). For \( x < 0 \), the output follows one rule, while for \( x > 0 \), it follows another, with a specific value at \( x = 0 \).

This structure makes piecewise functions versatile for modeling real-world scenarios where different rules apply in different conditions.

Limits

Limits are essential to understanding the behavior of functions as they approach a certain point. The limit of a function as \( x \) approaches a value \( a \) helps determine if the function is continuous at that point.

For a function \( f(x) \) to be continuous at \( x=a \), the limit as \( x \) approaches \( a \) must equal \( f(a) \). If it doesn't, the function is discontinuous at that point. Limits help describe what happens "close to" a point, even if the function isn't defined precisely at that point.

Understanding limits aids in identifying holes and jumps in the graph of a function, which are key in distinguishing between continuous and discontinuous functions.

Graphical Representation of Functions

Graphical representations provide a visual understanding of functions. When you plot a function's graph, you can immediately see its continuity or discontinuity.

For a continuous function like \( f(x) = 2x + 1 \), the graph is a straight line with no breaks. You can draw it without lifting your pen. Meanwhile, for a piecewise-defined function, you might have to draw separate lines or curves for different domains, possibly lifting your pen for discontinuities.

The graph is your tool to spot gaps, jumps, or holes. Open circles are commonly used to indicate points that are not part of the function, showing discontinuities.

Mathematical Continuity

Mathematical continuity is about unbroken behavior. A function is continuous over an interval if there are no gaps, jumps, or holes in its graph across that interval.

A function is continuous at a point \( x=a \) if it meets three conditions:

• \( f(a) \) is defined.
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( a \) equals \( f(a) \).

These criteria ensure that the function flows smoothly through \( x=a \) without interruptions.

Discontinuous functions, on the other hand, break these rules, leading to parts of their graphs that aren't connected.

Domain of a Function

The domain of a function is the complete set of input values (\( x \) values) for which the function is defined. Understanding the domain is crucial when analyzing continuity.

For instance, the linear function \( f(x) = 2x + 1 \) has a domain of all real numbers \((-\infty, \infty)\), making it continuous everywhere. Meanwhile, a piecewise function might have specific domains for each piece, like our earlier example where different rules apply for \( x < 0 \), \( x = 0 \), and \( x > 0 \).

Analyzing the domain helps see where a function may face discontinuities, especially at the borders between different definitions of a piecewise function.