Question

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\frac{|x-5|}{x-5} $$

Short answer

The function \(f(x)=\frac{|x-5|}{x-5}\) is not continuous at x=5 and has a removable discontinuity at this point.

Step-by-step solution

Understand the function

The function given is \(f(x)=\frac{|x-5|}{x-5}\). This is a piecewise function because the absolute value can be split into two cases: \(x-5\) when \(x\geq5\) and \(-(x-5)\) when \(x<5\).

Split the function

So when \(x\geq5\), \(f(x)=\frac{x-5}{x-5}=1\), and when \(x<5\), \(f(x)=\frac{-(x-5)}{x-5}=-1\). So now we have a simplified piecewise function where f(x)=1 when \(x\geq5\) and -=1 when \(x<5\).

Identify discontinuities

Having two different values of the function at x=5 suggests a discontinuity. Therefore, the function is not continuous at x=5.

Determine type of discontinuity

The discontinuity occurs because there is a specific x-value where the limits from the left and right are not equal. However, since the function does not tend towards infinity and the limit exists, this is a removable or 'point' discontinuity.

Key concepts

Removable Discontinuity

A removable discontinuity occurs when there is a hole in the graph of a function. This happens because the limit exists at that specific point, but the function either is not defined there or is defined differently. In other words, both sides of the function approach the same value, but the function doesn't reach this value at the point of interest.

In the function given as an example, we saw that as we approached from either side towards the point where the discontinuity exists, namely at \(x=5\), the limit from the left was equal to the limit from the right, but the function itself was not defined the same way across this point. Therefore, this creates a removable discontinuity at \(x=5\).

To "remove" the discontinuity, you can redefine the function at \(x=5\). For example, if \(f(x)\) was revisited such that \(f(5) = 1\), then the discontinuity would be removed, and the function would be continuous at that point.

Nonremovable Discontinuity

A nonremovable discontinuity, by contrast, is a type of discontinuity that cannot be eliminated by simply redefining a single point. These occur when there is a jump between two parts of the graph, an asymptote, or when the value of the function goes off towards infinity.

These types of discontinuities are typically observed when:

• The left-hand limit and right-hand limit do not equal each other.
• The function tends towards infinity at a certain point.
• There are vertical asymptotes.

In the given function example \(f(x)=\frac{|x-5|}{x-5}\), the discontinuity at \(x=5\) is not nonremovable because the limits from both sides exist and are equal, which means it doesn’t meet the criteria for a nonremovable discontinuity.

Piecewise Function

A piecewise function is made up of different segments or "pieces", each of which is defined by different expressions based on the input value. These functions are useful for modeling situations where a rule or pattern changes.

In our exercise, when looking at \(f(x)=\frac{|x-5|}{x-5}\), the absolute value function creates the necessity for a piecewise definition. This particular function can be expressed in two cases:

• \(f(x)=1\) for \(x \geq 5\)
• \(f(x)=-1\) for \(x < 5\)

Understanding piecewise functions allows us to grasp how different rules apply to different intervals and how they come together to form a complete graph. It highlights the points at which behaviors change, which are often points of interest when studying continuity and discontinuities.

Continuity

Continuity in a function implies there are no interruptions, jumps, or breaks at any point within its domain. Mathematically, a function is continuous at a point \(x = a\) if the limit of the function as it approaches from both the left and the right equals the function’s value at that point. More precisely:

• The function is defined at \(x = a\)
• The limit of the function as \(x\) approaches \(a\) exists
• The limit equals \(f(a)\)

In our example, \(f(x)=\frac{|x-5|}{x-5}\), we identified the function as being not continuous at \(x=5\). The limits from the left and right were equal, suggesting continuity criteria #2 was met, but the function itself was not defined the same all through at \(x=5\). Therefore, while the function showcased a concept of limits looking continuous, it failed the condition of having an uninterrupted value at \(x=5\). Understanding continuity helps to analyze functions for smooth behavior over their domains.