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? \ $$ f(x)=\left\\{\begin{array}{ll}{\tan \frac{\pi x}{4},} & {|x|<1} \\ {x,} & {|x| \geq 1}\end{array}\right. $$

Short answer

The function \(f(x)\) is discontinuous at \(x=-1\) and \(x=1\) and neither of these is a removable discontinuity.

Step-by-step solution

Identify where the function might be discontinuous

The function \(f\) might be discontinuous at any \(x\)-values where the function switches from one rule to another—that is, at \(x=-1\) and \(x=1\)—as well as at any \(x\)-values where either of the rules are undefined. For \(\tan(\pi x/4)\), this occurs whenever \(\pi x/4\) is an odd multiple of \(\pi/2\) because the tangent function has vertical asymptotes at these values.

Solve for x when the function changes and when tangent is undefined

We can see that the equation \(\pi x/4 = (2n+1)\pi/2\) (where \(n\) is any integer) is equivalent to \(x = 2(2n+1)\), so \(x\) is an odd multiple of 2. It can also be seen that \(-1<x<1\) only when \(x=-1,0,1\), and at these points, \(\tan(\pi x/4)\) is perfectly defined. Hence, \(f(x)\) is continuous for all \(|x|<1\). At \(x=-1\) and \(x=1\), the left-hand and right-hand limits are not the same, thus the function is discontinuous at these points.

Identify the type of the discontinuity

A removable discontinuity, also called a hole, occurs when the function is not defined at a particular \(x\)-value but the limit of the function as x approaches that value from both directions is the same. Both \(x=-1\) and \(x=1\) are not removable discontinuities since the left-hand and right-hand limits of the function at these points are different.

Key concepts

Removable Discontinuity

A removable discontinuity in a function is akin to a small gap or a 'hole' in the curve at a certain point. It happens when the function is technically undefined at a particular value of x, yet the left and the right limits as x approaches the value are equal. In simpler terms, if you were drawing the graph of the function, your pencil would lift off at that point, but then immediately continue from the direct opposite side of that gap.

It's possible to 'fix' this type of discontinuity. If we redefine the function value at this specific x to make it equal to the limit, we can 'remove' the discontinuity. This is why it's referred to as 'removable'. In algebraic terms, this often occurs when a factor in the numerator and denominator of a rational function cancels out, leaving a hole at that x-value.

Nonremovable Discontinuity

In contrast to removable discontinuities are nonremovable discontinuities, which represent more substantial breaks in the graph of a function. These occur when the function 'jumps' or has an asymptote — essentially, when the limits from the left and the right do not match or do not exist at all.

There are two common types of nonremovable discontinuities: jumps and infinite discontinuities. A jump occurs when the left and right-hand limits are finite but different. An infinite discontinuity, or asymptote, happens when at least one of the directional limits goes off towards infinity. In the given exercise, the function has a nonremovable discontinuity at x = -1 and x = 1 because it shifts from one formula to another, and the left and the right limits are divergent at these points.

Tangent Function

The tangent function, denoted as \(tan\), is one of the six fundamental trigonometric functions. It is particularly unique because its graph has a repeating pattern of vertical asymptotes. For any angle \(\theta\), tangent is defined as the ratio of the sine and the cosine of that angle: \(tan(\theta) = \frac{sin(\theta)}{cos(\theta)}\).

Because cosine equals zero at odd multiples of \(\frac{\pi}{2}\), the tangent function is undefined at these points, leading to the vertical asymptotes in its graph. This is why there are certain x-values where a function involving the tangent can be discontinuous, as is addressed in our exercise with \(tan(\frac{\pi x}{4})\).

Limits of a Function

Limits are a foundational concept in calculus that deal with the behavior of a function as the input approaches a certain value. They help us understand the function's values near a specific point, even if the function is not explicitly defined at that point.

The formal definition involves approaching the value of interest from both the left and the right side, denoted as the limit as x approaches 'c' from the left (\(lim_{x \to c^-} f(x)\)) and from the right (\(lim_{x \to c^+} f(x)\)). If both of these one-sided limits are equal, then the function has a limit at that point. In the case of the exercise provided, the limits do not match when approaching x = -1 and x = 1, highlighting the discontinuity of the function at these points.