Chapter 6: Problem 1

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

Short Answer

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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.

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

Short Answer

Step by step solution

1. Define Continuous and Discontinuous Functions

2. Example of a Continuous Function

3. Example of a Discontinuous Function

4. Draw Graphs for Both Types of Functions

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