Suppose g(x) = \frac{1}{x - 2} if x < 1, and 2x - 4 if x is greater than or equal to 1. The best...

Question:

Suppose {eq}\; g(x) = \left\{\begin{matrix} \displaystyle\frac{1}{x - 2} & \textrm{ if } x < 1 \\ 2x - 4 **** \textrm{ if } x \geq 1 \end{matrix}\right. {/eq}

The best description concerning the continuity of {eq}g(x) {/eq} is that the function

A.) {eq}\textrm{ is continuous.} {/eq}

B.) {eq}\textrm{ has a jump discontinuity.} {/eq}

C.) {eq}\textrm{ has an infinite discontinuity.} {/eq}

D.) {eq}\textrm{ has a removable discontinuity.} {/eq}

E.) {eq}\textrm{ None of the above.} {/eq}

Continuity:

We are given a function and asked to determine the continuity. To do this, we have to make sure we account for every point and inspect any possible discontinuities. This is true for piecewise-defined functions in particular since even if the functions that are given are continuous, discontinuities could exist at the points where the function changes definition.

Answer and Explanation:

We are asked to inspect the continuity of the function $$g(x) = \left\{ \begin{array}{rl} \dfrac{1}{x - 2} & \text{if } x < 1 \\ 2x - 4 & \text{if } x \geq 1 \end{array} \right. $$

Note that for the given intervals, each of the functions in the definition is continuous because {eq}\dfrac{1}{x - 2} {/eq} is continuous everywhere except {eq}x = 2 {/eq} which is not contained in the interval where this function is given. Also, {eq}2x - 4 {/eq} is a polynomial so it is continuous everywhere. The only possible discontinuity is where the function changes its definition, which is at {eq}x = 1 {/eq}.

We can find the limit from both sides to determine the continuity at this point. We have $$\begin{align*} \lim_{x \to 1^-}{g(x)} &= \lim_{x \to 1^-}{\frac{1}{x - 2}} \\ &= -1 \end{align*} $$ and $$\begin{align*} \lim_{x \to 1^+}{g(x)} &= \lim_{x \to 1^+}{(2x - 4)} \\ &= -2 \end{align*} $$ and since the limits are different, we can say that {eq}g {/eq} has a jump discontinuity.

Therefore the answer is B.


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One-Sided Limits and Continuity

from Math 104: Calculus

Chapter 6 / Lesson 3
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