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.