Discontinuity of Trigonometric Functions

Trigonometric functions are an important type of function in calculus, and some of them are discontinuous at certain points.

Understanding the discontinuity of trigonometric functions is essential for mastering the concept of discontinuity.

Basic Properties

Trigonometric functions have the following basic properties:

• Periodicity: have periodic features
• Continuity: $\sin x$ and $\cos x$ are continuous on $\mathbb{R}$
• Discontinuity: $\tan x$ and $\cot x$ are discontinuous at certain points
• Asymptotes: there is a vertical asymptote at the points of discontinuity

Continuous Trigonometric Functions

The Sine and Cosine Functions

Sine function: $f(x) = \sin x$

Properties:

• Domain: $\mathbb{R}$
• Continuous everywhere on $\mathbb{R}$
• Period $2\pi$
• Range: $[-1, 1]$

Cosine function: $f(x) = \cos x$

Properties:

• Domain: $\mathbb{R}$
• Continuous everywhere on $\mathbb{R}$
• Period $2\pi$
• Range: $[-1, 1]$

Discontinuous Trigonometric Functions

The Tangent Function

Properties:

• Domain: $x \neq \frac{\pi}{2} + k\pi$ ($k$ an integer)
• Continuous on its domain
• Period $\pi$
• Range: $\mathbb{R}$
• Has a vertical asymptote at every point of discontinuity

Analysis of the points of discontinuity:

• Points of discontinuity: $x = \frac{\pi}{2} + k\pi$ ($k$ an integer)
• Left limit: $\lim_{x \to (\frac{\pi}{2} + k\pi)^-} \tan x = +\infty$
• Right limit: $\lim_{x \to (\frac{\pi}{2} + k\pi)^+} \tan x = -\infty$
• These are discontinuities of the second kind

The Cotangent Function

Properties:

• Domain: $x \neq k\pi$ ($k$ an integer)
• Continuous on its domain
• Period $\pi$
• Range: $\mathbb{R}$
• Has a vertical asymptote at every point of discontinuity

Analysis of the points of discontinuity:

• Points of discontinuity: $x = k\pi$ ($k$ an integer)
• Left limit: $\lim_{x \to (k\pi)^-} \cot x = -\infty$
• Right limit: $\lim_{x \to (k\pi)^+} \cot x = +\infty$
• These are discontinuities of the second kind

The Secant and Cosecant Functions

Secant function: $f(x) = \sec x = \frac{1}{\cos x}$

Properties:

• Domain: $x \neq \frac{\pi}{2} + k\pi$ ($k$ an integer)
• Continuous on its domain
• Period $2\pi$
• Has a vertical asymptote at every point of discontinuity

Cosecant function: $f(x) = \csc x = \frac{1}{\sin x}$

Properties:

• Domain: $x \neq k\pi$ ($k$ an integer)
• Continuous on its domain
• Period $2\pi$
• Has a vertical asymptote at every point of discontinuity

Analysis of the Discontinuity of Trigonometric Functions

Types of Point of Discontinuity

The points of discontinuity of trigonometric functions mainly fall into the following types:

1. Discontinuities of the second kind: the limit tends to infinity
2. Vertical asymptotes: there is a vertical asymptote at the point of discontinuity
3. Periodic discontinuities: the points of discontinuity have periodic features

Determining Continuity

1. $\sin x$ and $\cos x$: continuous everywhere on $\mathbb{R}$
2. $\tan x$ and $\cot x$: continuous on their domains, with a vertical asymptote at every point of discontinuity
3. $\sec x$ and $\csc x$: continuous on their domains, with a vertical asymptote at every point of discontinuity

Discontinuity of Composite Trigonometric Functions

Basic Theorem

If the function $f(x)$ is continuous at $x_0$, then $\sin(f(x))$ and $\cos(f(x))$ are continuous at $x_0$.

Examples

Example 1: $f(x) = \sin(\frac{1}{x})$

Analysis:

• The inner function $h(x) = \frac{1}{x}$ is continuous at $x \neq 0$
• The outer function $g(x) = \sin x$ is continuous on $\mathbb{R}$
• Therefore $f(x) = g(h(x))$ is continuous at $x \neq 0$
• It is discontinuous at $x = 0$ (the inner function is undefined at that point)

Example 2: $f(x) = \tan(x^2)$

Analysis:

• The inner function $h(x) = x^2$ is continuous on $\mathbb{R}$
• The outer function $g(x) = \tan x$ is continuous on its domain
• Therefore $f(x) = g(h(x))$ is continuous on its domain
• Points of discontinuity: those where $x^2 = \frac{\pi}{2} + k\pi$, that is, $x = \pm\sqrt{\frac{\pi}{2} + k\pi}$

Graph Characteristics of Trigonometric Functions

Summary of Graph Features

1. Periodicity: trigonometric functions are periodic, so the graph repeats itself
2. Smoothness: on its domain, the graph of a trigonometric function is a smooth curve
3. Asymptotes: there is a vertical asymptote at every point of discontinuity
4. Symmetry: $\sin x$ is an odd function and $\cos x$ is an even function

Exercises

Exercise 1

Determine the continuity of $f(x) = \tan(x^2)$ at $x = 0$.

Exercise 2

Determine the continuity of $f(x) = \sin(\frac{1}{x})$ at $x = 0$.

Exercise 3

Determine the intervals of continuity of $f(x) = \cot(x^2)$.

Exercise 4

Determine the continuity of $f(x) = \tan(\sin x)$ on $\mathbb{R}$.

Exercise 5

Determine the continuity of $f(x) = \cos(\ln x)$ at $x = 1$.

Exercise 6

Let $f(x) = \begin{cases} \sin x, & x \leq 0 \\ x + 1, & x > 0 \end{cases}$. Determine the continuity of $f(x)$ at $x = 0$.

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Summary

Symbols Used in This Article