Removable Discontinuity (Hole)
A removable discontinuity, often called a hole, is a point where the function is undefined, but the limit of the function as x approaches the point of discontinuity exists. For example, consider the function f(x) = x^2 for x ≠ 0 and undefined for x = 0. This function has a removable discontinuity at x = 0. The graph of this function will have a hole at x = 0. As x approaches 0 from either the left or the right, the function approaches the same value, but the function is not defined at x = 0. The hole, represented by an empty circle, signifies this missing point.
Jump Discontinuity
A jump discontinuity occurs when the function jumps from one value to another as it crosses a certain point. For example, consider the function f(x) = x for x < 0 and f(x) = 2 for x > 0. This function has a jump discontinuity at x = 0. The graph of this function will show a jump at x = 0. As x approaches 0 from the left, the function approaches 0, and as x approaches 0 from the right, the function approaches 2. The empty circle at x = 0 for f(x) = x and the filled circle at x = 0 for f(x) = 2 visually represent this jump.
Essential Discontinuity (Asymptote)
An essential discontinuity, often called an infinite discontinuity, occurs when the function’s value approaches infinity as x approaches a certain point. For example, the function f(x) = 1/x has an essential discontinuity at x = 0. The graph of this function will have a vertical asymptote at x = 0. As x approaches 0 from the left, the function approaches negative infinity, and as x approaches 0 from the right, the function approaches positive infinity. The vertical asymptote represents the x-value where the function goes to infinity.
Removable Discontinuity (Hole)
A removable discontinuity, often called a hole, is a point where the function is undefined, but the limit of the function as x approaches the point of discontinuity exists. For example, consider the function f(x) = x^2 for x ≠ 0 and undefined for x = 0. This function has a removable discontinuity at x = 0. The graph of this function will have a hole at x = 0. As x approaches 0 from either the left or the right, the function approaches the same value, but the function is not defined at x = 0. The hole, represented by an empty circle, signifies this missing point.
Jump Discontinuity
A jump discontinuity occurs when the function jumps from one value to another as it crosses a certain point. For example, consider the function f(x) = x for x < 0 and f(x) = 2 for x > 0. This function has a jump discontinuity at x = 0. The graph of this function will show a jump at x = 0. As x approaches 0 from the left, the function approaches 0, and as x approaches 0 from the right, the function approaches 2. The empty circle at x = 0 for f(x) = x and the filled circle at x = 0 for f(x) = 2 visually represent this jump.
Essential Discontinuity (Asymptote)
An essential discontinuity, often called an infinite discontinuity, occurs when the function’s value approaches infinity as x approaches a certain point. For example, the function f(x) = 1/x has an essential discontinuity at x = 0. The graph of this function will have a vertical asymptote at x = 0. As x approaches 0 from the left, the function approaches negative infinity, and as x approaches 0 from the right, the function approaches positive infinity. The vertical asymptote represents the x-value where the function goes to infinity.
Removable Discontinuity (Hole)
A removable discontinuity, often called a hole, is a point where the function is undefined, but the limit of the function as x approaches the point of discontinuity exists. For example, consider the function f(x) = x^2 for x ≠ 0 and undefined for x = 0. This function has a removable discontinuity at x = 0. The graph of this function will have a hole at x = 0. As x approaches 0 from either the left or the right, the function approaches the same value, but the function is not defined at x = 0. The hole, represented by an empty circle, signifies this missing point.
Jump Discontinuity
A jump discontinuity occurs when the function jumps from one value to another as it crosses a certain point. For example, consider the function f(x) = x for x < 0 and f(x) = 2 for x > 0. This function has a jump discontinuity at x = 0. The graph of this function will show a jump at x = 0. As x approaches 0 from the left, the function approaches 0, and as x approaches 0 from the right, the function approaches 2. The empty circle at x = 0 for f(x) = x and the filled circle at x = 0 for f(x) = 2 visually represent this jump.
Essential Discontinuity (Asymptote)
An essential discontinuity, often called an infinite discontinuity, occurs when the function’s value approaches infinity as x approaches a certain point. For example, the function f(x) = 1/x has an essential discontinuity at x = 0. The graph of this function will have a vertical asymptote at x = 0. As x approaches 0 from the left, the function approaches negative infinity, and as x approaches 0 from the right, the function approaches positive infinity. The vertical asymptote represents the x-value where the function goes to infinity.
This is a graph of the function \(y = 5\) with a removable discontinuity (a hole) at \(x = -1\) and a jump discontinuity at \(x = 2\).
At \(x = -1\), there is a removable discontinuity, also known as a hole. This means that the function is not defined at \(x = -1\), but if we were to fill in this point, the function would be continuous.
At \(x = 2\), there is a jump discontinuity. The function jumps from \(y = 5\) to \(y = 0\). The white circle at \(x = 2, y = 5\) indicates the excluded endpoint, and the black dot at \(x = 2, y = 0\) indicates the included part.
This graph represents a piecewise function with two types of discontinuities. The function is defined as \(1/(x+1)^2\) for \(x < -1\) and \(-1 < x < 2\), and as a constant \(3\) for \(x > 2\).
At \(x = -1\), there is a vertical asymptote. This is represented by the dashed vertical line. The function approaches infinity as \(x\) approaches -1, which is characteristic of a vertical asymptote.
At \(x = 2\), there is a jump discontinuity. The function jumps from the value of \(1/(2+1)^2\) which simplifies to \(1/9\), to the constant value of \(3\). This is represented by the hole at the point \((2, 1/9)\) and the solid dot at the point (2,3).
The dashed vertical line at \(x = -1\) and the hole and solid dot at \(x = 2\) clearly illustrate the discontinuities in the function.
This graph represents a piecewise function with two types of discontinuities. The function is defined as \(1/(x+1)^2\) for all \(x\) except at \(x = -1\) and \(x = 2\).
At \(x = -1\), there is a vertical asymptote. This is represented by the dashed vertical line. The function approaches infinity as \(x\) approaches -1, which is characteristic of a vertical asymptote.
At \(x = 2\), there is a removable discontinuity, also known as a hole. This is not because the function is undefined at \(x = 2\), but because we’ve chosen to place a hole there. This is represented by the hole at the point \((2, 1/9)\).
The dashed vertical line at \(x = -1\) and the hole at \(x = 2\) clearly illustrate the discontinuities in the function.
Understanding the Piecewise Function
Let’s take a closer look at the piecewise function defined as \(y = x^2\) for \(x \leq -1\) and \(y = 3\) for \(x > -1\).
As we move from left to right across the graph:
- For \(x \leq -1\), the function behaves as \(y = x^2\), decreasing as \(x\) approaches -1 from the left. The solid dot at \((-1,1)\) indicates that this point is included in the function.
- At \(x = -1\), there is a discontinuity. The function jumps from \((-1,1)\) to \((-1,3)\). The hollow circle at \((-1,3)\) indicates that this point is not included in the function.
- For \(x > -1\), the function is constant at \(y = 3\).
This behavior is characteristic of piecewise functions, which can have different forms and properties in different parts of their domain.
Understanding the Function with a Hole
Let’s take a closer look at the function defined as \(y = x^2 + 1\) with a hole at \(x = 2\).
As we move from left to right across the graph:
- For \(x < 2\), the function behaves as \(y = x^2 + 1\), increasing as \(x\) approaches 2.
- At \(x = 2\), there is a hole in the graph. The function is not defined at this point.
- For \(x > 2\), the function continues as \(y = x^2 + 1\), increasing as \(x\) moves away from 2.
This behavior is characteristic of functions with a point of discontinuity, where the function is not defined.
Understanding the Function with a Hole and a Defined Point
Let’s take a closer look at the function defined as \(y = x^2 + 1\) with a hole at \(x = 2\) and a solid dot at the point (2,2).
As we move from left to right across the graph:
- For \(x < 2\), the function behaves as \(y = x^2 + 1\), increasing as \(x\) approaches 2.
- At \(x = 2\), there is a hole in the graph. The function is not defined at this point in the original function. However, the solid dot at the point (2,2) indicates that the function is defined at \(x = 2\) with a value of 2.
- For \(x > 2\), the function continues as \(y = x^2 + 1\), increasing as \(x\) moves away from 2.
This behavior is characteristic of functions with a point of discontinuity, where the function is not defined, but a value is assigned at that point.