Problem 20 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(19-24,(\) a) find each point of discontinuity. (b) Which of the discontinuities are removable? not removable? Give reasons for your answers. $$f(x)=\left\\{\begin{array}{ll}{3-x,} & {x<2} \\ {2,} & {x=2} \\ {x / 2,} & {x>2}\end{array}\right.$$

Step-by-Step Solution

Answer

The function \(f(x)\) is discontinuous at \(x=2\). This discontinuity is removable.

Step 1: Identifying discontinuities

First, let's identify the x-values where there are potential discontinuities. In this case, because the function switches from one expression to another at x=2, we need to check at the point \(x=2\). For \(x<2\), the function is defined by \(3-x\), and for \(x>2\) the function is defined by \(x/2\). Therefore, we must evaluate the limits of the function at the point \(x=2\) for each of these ranges.

Step 2: Calculate limits

The limits for \(x<2\) and \(x>2\) can be evaluated as follows:\[\lim_{x \to 2^-} f(x) = 3 - 2 = 1\]\[\lim_{x \to 2^+} f(x) = 2/2 = 1.\]Since these limits are equal, there is no jump or break in the graph at \(x=2\). However, \(f(2) = 2\), which does not equal to the limit at \(x=2\), therefore, \(f(x)\) is discontinuous at \(x=2\).

Step 3: Classifying discontinuity as removable or not removable

A discontinuity is said to be removable if the function can be redefined at the point of discontinuity so that the new function is continuous at that point. Here, discontinuity at \(x=2\) can be removed by redefining \(f(2)\) to be 1 (the limit at \(x=2\)), instead of 2. The function with new definition would then be continuous at \(x=2\). Hence, the discontinuity at \(x=2\) is removable.

Key Concepts

Removable Discontinuity

When studying the behavior of functions, it's important to recognize different types of discontinuities. A removable discontinuity occurs at a point on a graph where a function is not defined or does not match the limiting value, but where the limit nonetheless exists. Such a discontinuity can be 'fixed' by redefining the function at that point to equal the limit.

Consider the provided function where a discrepancy is noted at the point where x equals 2. The limits from either side of the point are the same, indicating that the function approaches a single, common value, but the function itself gives a different value at exactly x=2. By redefining the function value at x=2 to match the limits, we can make it continuous, effectively 'removing' the discontinuity. This adjustment is a clear illustration of when a discontinuity is removable, emphasizing the concept's significance in creating smoother, more predictable functions.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the value that a function approaches as the input approaches a certain point. Mathematically, limits help us address function behavior near a point, not necessarily at that point itself.

In our example, the limits were calculated as the input x approached 2 from both the left (\(x \to 2^-\)) and the right (\(x \to 2^+\)), resulting in a limit of 1 from both sides. This consistency of limits is crucial for determining the existence and type of discontinuity. If the limits had not matched, we would have faced a different kind of discontinuity, such as a jump discontinuity. Understanding limits is therefore not only about finding a function’s behavior at specific points but also about discerning the overall continuity of the function.

Continuous Function

A continuous function is one where small changes to the input result in small changes to the output. More formally, this implies that for every point within a function’s domain, the limit of the function as x approaches that point is equal to the function's value at that point.

The concept of continuity is essential because continuous functions are predictable and can be graphed without lifting your pencil from the paper. The example function given in the exercise was continuous except at x=2, where a single, isolated point of discontinuity occurred – a removable discontinuity. It's important to recognize and understand such disruptions in continuity, as many theorems and calculations in calculus rely on functions being continuous over certain intervals.