https://file.aiquickdraw.com/mathbot/file/17289970548349r50fgb1.jpg

The question asks: "Which functions are discontinuous on \(\mathbb{R}\) among the following?"

The options listed are:
1. \(\sinh x\)
2. \(\cos x\)
3. \(\cosh x\)
4. \(\cot x\)
5. \(\tan x\)
6. \(\sin x\)

Let's analyze them:

- **\(\sinh x\)**: The hyperbolic sine function is continuous everywhere on \(\mathbb{R}\).
- **\(\cos x\)**: The cosine function is continuous everywhere on \(\mathbb{R}\).
- **\(\cosh x\)**: The hyperbolic cosine function is also continuous everywhere on \(\mathbb{R}\).
- **\(\cot x\)**: The cotangent function, \(\cot x = \frac{1}{\tan x}\), is discontinuous at points where \(\tan x = 0\), which are multiples of \(\pi\), such as \(x = 0, \pi, 2\pi, \dots\). So, this function is discontinuous.
- **\(\tan x\)**: The tangent function is discontinuous where \(\cos x = 0\), which occurs at odd multiples of \(\frac{\pi}{2}\), such as \(x = \frac{\pi}{2}, \frac{3\pi}{2}, \dots\). Hence, this function is also discontinuous.
- **\(\sin x\)**: The sine function is continuous everywhere on \(\mathbb{R}\).

### Conclusion:
The correct answers are **\(\cot x\)** and **\(\tan x\)**, which are the ones you have selected in the image. These are the functions that are discontinuous on \(\mathbb{R}\).

Would you like more details or have any questions?

### 5 Related Questions:
1. Why is \(\cot x\) discontinuous at multiples of \(\pi\)?
2. How can we identify points of discontinuity for trigonometric functions in general?
3. What is the difference between the continuity of hyperbolic and trigonometric functions?
4. Can you explain why \(\tan x\) is undefined at odd multiples of \(\frac{\pi}{2}\)?
5. What are the graphs of \(\tan x\) and \(\cot x\) near their discontinuities?

### Tip:
To check the continuity of a function, look for points where its denominator equals zero (if it's a rational function) or where the function is not defined.

Đâu là hàm số không liên tục trên R trong các hàm dưới đây?

Identify the discontinuities in trigonometric functions like cotangent and tangent on the real number line. Learn why cot(x) and tan(x) are discontinuous and how to determine their points of discontinuity. Suitable for high school students studying advanced trigonometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation