Problem 7
Question
Sketch the graph of a function \( f \) that is continuous except for the stated discontinuity. Removable discontinuity at 3, jump discontinuity at 5.
Step-by-Step Solution
Verified Answer
Draw a line with a hole at 3 and a break at 5 with a vertical offset.
1Step 1: Understanding Discontinuities
A removable discontinuity at a point occurs when a function is not defined at that point, but the limit exists. For a jump discontinuity, the left-hand limit and the right-hand limit exist, but they are not equal.
2Step 2: Start with a Continuous Function
Begin by sketching a continuous function, such as a simple linear function, across the domain. For simplicity, we'll use the line \( y = x \).
3Step 3: Add Removable Discontinuity at 3
To create a removable discontinuity at \( x = 3 \), erase the point where the function meets it, making it look as though there's a hole at \( x = 3 \) on the graph. The graph should continue smoothly on either side of the hole.
4Step 4: Add Jump Discontinuity at 5
Introduce a jump discontinuity at \( x = 5 \). To do this, decide the height of the jump. For instance, keep the function at \( y = x \) until just before \( x = 5 \), then suddenly jump to \( y = x + 2 \) starting at \( x = 5 \) for all \( x > 5 \). This represents the jump discontinuity.
5Step 5: Final Sketch
On the graph, indicate a continuous line that passes through all points of the domain except at \( x = 3 \) for the removable discontinuity, where it should be an empty circle, and a gap between the line segments at \( x = 5 \) for the jump discontinuity.
Key Concepts
Removable DiscontinuityJump DiscontinuityContinuous Functions
Removable Discontinuity
Removable discontinuities are a fascinating concept in graph sketching. These occur when a function has a missing point at a particular location, but the behavior of the graph around this point is well-behaved. In simpler terms, you can think of it as a hole in the graph. Just like a donut hole doesn't affect the round shape of the donut, a removable discontinuity doesn't disrupt the overall graph structure.
In mathematical terms, a removable discontinuity at a point, say at 3 as in our exercise, means that while the function, let's call it \( f \), is not defined at \( x = 3 \), the limit as \( x \) approaches 3 exists. This suggests that the gap or hole could technically be 'fixed' or filled by defining \( f(3) \) to be whatever the limit is. However, in exercises like these, we're sketching what the function behaves like when left unfixed – imagine erasing a dot at \( x = 3 \) on the line graph and continuing the line smoothly on either side of this hole.
In mathematical terms, a removable discontinuity at a point, say at 3 as in our exercise, means that while the function, let's call it \( f \), is not defined at \( x = 3 \), the limit as \( x \) approaches 3 exists. This suggests that the gap or hole could technically be 'fixed' or filled by defining \( f(3) \) to be whatever the limit is. However, in exercises like these, we're sketching what the function behaves like when left unfixed – imagine erasing a dot at \( x = 3 \) on the line graph and continuing the line smoothly on either side of this hole.
- Occurs at a single point where the function isn't defined.
- Limits from both sides exists and are equal.
- Graphically represented as a 'hole' in the graph, not affecting continuity elsewhere.
Jump Discontinuity
Jump discontinuity takes the idea of discontinuity into another interesting realm, as it's more abrupt and pronounced than a removable discontinuity. In this kind of discontinuity, the graphed function suddenly 'jumps' from one value to another, creating a visible cleft in the graph.
Think of a jump discontinuity like a cliff on your hiking trail. As you reach the edge of the cliff, your path doesn't continue straight but instead drops or rises drastically at a certain point. For example, as given in our exercise, at \( x = 5 \), the path leaps from the current line at \( y = x \) to a height of \( y = x + 2 \), causing the 'jump'. Here, even though both left-hand and right-hand limits exist, they aren't equal, which gives the discontinuity its name and characteristic gap.
Think of a jump discontinuity like a cliff on your hiking trail. As you reach the edge of the cliff, your path doesn't continue straight but instead drops or rises drastically at a certain point. For example, as given in our exercise, at \( x = 5 \), the path leaps from the current line at \( y = x \) to a height of \( y = x + 2 \), causing the 'jump'. Here, even though both left-hand and right-hand limits exist, they aren't equal, which gives the discontinuity its name and characteristic gap.
- Occurs at a specific point where the function shifts abruptly.
- Different functional value on either side leads to a visible gap.
- Left and right-hand limits exist but are not equal.
- Graphically, it represents a 'jump' or gap within the plotted graph line.
Continuous Functions
Continuous functions are the smooth operators in the world of math graphs. They create unbroken, undisturbed paths over their domain, akin to a perfectly spun thread with no breaks. For a function to be truly continuous, it must satisfy a few key criteria:
The primary characteristic of a continuous function, let's say our line \( y = x \), is that you can draw it without lifting your pencil from the paper. There are no gaps, jumps, or holes in its line. The intuitive idea is that small changes in \( x \) bring small changes in \( y \), which can be confirmed mathematically when the limit of \( f(x) \) as \( x \) approaches any point in the function's domain equals \( f(x) \) itself. This property is precisely what makes functions like \( y = x \) ideal for beginning a graph before introducing discontinuities.
The primary characteristic of a continuous function, let's say our line \( y = x \), is that you can draw it without lifting your pencil from the paper. There are no gaps, jumps, or holes in its line. The intuitive idea is that small changes in \( x \) bring small changes in \( y \), which can be confirmed mathematically when the limit of \( f(x) \) as \( x \) approaches any point in the function's domain equals \( f(x) \) itself. This property is precisely what makes functions like \( y = x \) ideal for beginning a graph before introducing discontinuities.
- Smooth graph with no breaks, jumps, or holes.
- The limit of \( f(x) \) exists and equals \( f(x) \) at every point in the domain.
- Can be sketched without lifting your pen or pencil from the paper.
Other exercises in this chapter
Problem 7
Find an equation of the tangent line to the curve at the given point. \( y = \sqrt{x} \), \( (1, 1) \)
View solution Problem 7
Sketch the graph of an example of a function \( f \) that satisfies all of the given conditions. \( \displaystyle \lim_{x \to 2} f(x) = -\infty \), \( \displays
View solution Problem 7
For the limit $$ \lim_{x \to 2}(x^3 - 3x + 4) = 6 $$ illustrate Definition 2 by finding values of \( \delta \) that correspond to \( \varepsilon = 0.2 \) and \(
View solution Problem 7
Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). \( \displaystyle \lim_{x \to 8}(1 + \sqrt[3]{x})(2 - 6x^2 + x^3) \)
View solution