Question Statement

Determine whether the function $f(x)=\frac{1}{x}$ is continuous on the following intervals:

(a) $(-3,3)$

(b) $(0,10]$

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Background and Explanation

A function is continuous on an interval if it is defined at every point within that interval. The function $f(x)=\frac{1}{x}$ is a rational function with a vertical asymptote at $x=0$, where it is undefined.

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Solution

Part (a): Interval $(-3,3)$

The function $f(x)=\frac{1}{x}$ is not defined at $x=0$.

Since $0\in (-3,3)$, the point where the function is undefined lies within this interval.

$\Rightarrow f(x)\text{ is discontinuous in interval }(-3,3).$

Part (b): Interval $(0,10]$

As $x=0\notin (0,10]$, the function is defined for every point in this interval.

Since $f(x)=\frac{1}{x}$ is continuous at all points in its domain (all real numbers except $0$), and the entire interval $(0,10]$ lies within the domain:

$f(x)\text{ is continuous in interval }(0,10].$

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Key Formulas or Methods Used

• Continuity on an interval: A function is continuous on an interval $I$ if and only if it is continuous at every point $c\in I$
• Domain of rational functions: $f(x)=\frac{1}{x}$ is undefined at $x=0$
• Set membership test: Checking whether $0$ belongs to the open interval $(-3,3)$ or the half-open interval $(0,10]$

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Summary of Steps

1. Identify the point where $f(x)=\frac{1}{x}$ is undefined ($x=0$)
2. For interval $(-3,3)$: Check if $0$ is contained in the interval; since $0\in (-3,3)$, the function is discontinuous
3. For interval $(0,10]$: Check if $0$ is excluded from the interval; since $0\notin (0,10]$, the function is continuous throughout