Question Statement

Determine the continuity of the function $f(x)=\sin \frac{1}{x}$ on the following intervals:

(a) $\left[\frac{1}{\pi },5\right)$

(b) $\left[\frac{\pi }{2},\frac{3\pi }{2}\right]$

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Background and Explanation

The function $f(x)=\sin \frac{1}{x}$ is a composition of the sine function and the reciprocal function. Since $\sin (u)$ is continuous everywhere and $\frac{1}{x}$ is continuous for all $x\ne 0$, the composition is continuous on any interval that does not contain $x=0$.

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Solution

Part (a)

Consider the interval $\left[\frac{1}{\pi },5\right)$.

Since $f(x)=\sin \frac{1}{x}$ is defined for all $x\in \left[\frac{1}{\pi },5\right]$ (as $x\ge \frac{1}{\pi }>0$ ensures $x\ne 0$), and both the reciprocal function and sine function are continuous on their respective domains, their composition is continuous.

$\Rightarrow f(x)\text{ is continuous in interval }\left[\frac{1}{\pi },5\right)$

Part (b)

Consider the interval $\left[\frac{\pi }{2},\frac{3\pi }{2}\right]$.

Since $f(x)$ is defined for all $x\in \left[\frac{\pi }{2},\frac{3\pi }{2}\right]$ (note that $\frac{\pi }{2}\approx 1.57>0$, so $x\ne 0$ throughout this interval), the composition remains continuous.

$\Rightarrow f(x)\text{ is continuous in interval }\left[\frac{\pi }{2},\frac{3\pi }{2}\right]$

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Key Formulas or Methods Used

• Continuity of Composition: If $g$ is continuous at $x=a$ and $h$ is continuous at $g(a)$, then $(h\circ g)(x)=h(g(x))$ is continuous at $x=a$
• Domain of Reciprocal Function: $\frac{1}{x}$ is continuous and defined for all $x\in \mathbb{R}\setminus \{0\}$
• Continuity of Sine: $\sin (x)$ is continuous for all $x\in \mathbb{R}$

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Summary of Steps

1. Identify the function structure: Recognize $f(x)=\sin \frac{1}{x}$ as a composition of $h(u)=\sin (u)$ and $g(x)=\frac{1}{x}$
2. Check domain restrictions: Verify that $x\ne 0$ for all $x$ in the given intervals
3. Apply continuity rules: Conclude that since both component functions are continuous on the relevant domains, the composition is continuous
4. For interval (a): Confirm $\left[\frac{1}{\pi },5\right)$ does not contain $0$ (since $\frac{1}{\pi }>0$), hence $f$ is continuous
5. For interval (b): Confirm $\left[\frac{\pi }{2},\frac{3\pi }{2}\right]$ does not contain $0$ (since $\frac{\pi }{2}>0$), hence $f$ is continuous