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? \ $$ f(x)=\left\\{\begin{array}{ll}{\frac{1}{2} x+1,} & {x \leq 2} \\ {3-x} & {x>2}\end{array}\right. $$

Short answer

The function \(f(x)\) is discontinuous at \(x=2\). The type of discontinuity is a jump discontinuity, which is not removable.

Step-by-step solution

Identify the point of possible discontinuity

The point of possible discontinuity would be the point where our function definition changes. Here, it changes at \(x=2\).

Evaluate the limit of the function as it approaches the possible discontinuity from the left and from the right

The left limit, denoted as \( \lim_{x \to 2^-} f(x) \), is the limit of the function as \(x\) approaches 2 from the left. We use the first piece of our function to calculate: \( \lim_{x \to 2^-} f(x) = \frac{1}{2}(2) + 1 = 2\). For the right limit, denoted as \( \lim_{x \to 2^+} f(x) \), we use the second piece of our function: \( \lim_{x \to 2^+} f(x) = 3 - 2 = 1\).

Compare the left and right limits

If the left limit and right limit are equal, then the function is continuous at \(x = 2\). Here, the left limit is 2 and the right limit is 1, so they are not equal. Hence, the function is not continuous at \(x=2\).

Determine the type of discontinuity

Since our function is defined at \(x = 2\), we have a jump discontinuity, which is not removable. If the function had not been defined at all at \(x = 2\), there would have been a hole, which is a removable discontinuity.

Key concepts

Removable Discontinuity

When a function 'jumps' from one value to another at a particular point, there's a good chance we're dealing with a form of discontinuity. Specifically, a removable discontinuity exists when a point on the graph is undefined or not congruent with the rest of the function's values, but can be 'fixed' by redefining the function at that point. Imagine a small hole in the curve of a graph that can be filled with a single, precise value.

Imagine the function \( f(x) = (x^2 - 1)/(x - 1) \). At \( x = 1 \), the function isn't defined since it results in a zero denominator. However, because \( f(x) \) can be simplified to \( x + 1 \) (by factoring and cancelling out \( (x-1) \)), and since \( x + 1 \) is defined at \( x = 1 \) (where \( f(x) \) was not), it demonstrates a removable discontinuity at this point. With limits, if \( \lim_{x\to c^-} f(x) = \lim_{x\to c^+} f(x) = L \) but \( f(c) \) is not equal to \( L \) or is not defined, then \( c \) is the point of removable discontinuity.

Nonremovable Discontinuity

Conversely, a nonremovable discontinuity is like a break or leap in the graph that cannot be 'repaired' by simply redefining the function at that point. Think of it as a gap or jump that's too large to fix with a single point. This kind of discontinuity usually indicates a fundamental change in the behaviour or structure of the function.

For instance, our exercise shows a nonremovable discontinuity for the function \( f(x) \). At \( x = 2 \), the limits from the left and right aren't equal which points to an abrupt change. This sudden 'jump' in values - from \( \lim_{x\to 2^-} f(x) = 2 \) to \( \lim_{x\to 2^+} f(x) = 1 \) - signifies a jump discontinuity, which clearly can’t be smoothed over by redefining \( f(2) \).

Limits

The concept of limits is pivotal in understanding both continuity and discontinuity in functions. A limit describes the value that a function approaches as the input (or 'x' value) approaches some value. They are essential for analyzing function behaviour and conforming to the function's trends without necessarily reaching a specific point on the function.

In the given exercise, evaluating limits from the left and right sides of \( x = 2 \) is crucial to determining the function's continuity. We say \( \lim_{x\to c} f(x) = L \) if the values of \( f(x) \) get arbitrarily close to \( L \) as \( x \) approaches \( c \) from either side. This is what we did in our steps, and the conflicting results confirmed the discontinuity.

Continuity

Lastly, continuity is what we have when there are no 'holes', 'jumps', or 'breaks' in a function. For a function to be considered continuous at a point \( x = c \) the following must be satisfied: \( f(c) \) is defined, \( \lim_{x\to c} f(x) \) exists, and \( \lim_{x\to c} f(x) = f(c) \). If any of these conditions fail, the function is discontinuous at that point.

Reflecting on our example, since the limit of \( f(x) \) as \( x \) approaches 2 from the left does not equal to the limit from the right, nor to the function's value at 2, the function breaks continuity at this juncture. A continuous function allows us to trace the graph without lifting the pencil, which wouldn’t be possible with the function \( f(x) \) at \( x=2 \).