Define a discontinuous sine function?

In summary, a function can only be continuous or discontinuous at points where it is defined, and the value at those points must be different. To create a function that is discontinuous at multiples of $\pi$, the definition should state that the function is equal to the sine function for all values of $x$ except for multiples of $\pi$, where it can be defined as any other value. The conditions in the two branches should be mutually exclusive, and the vertical bar should only be used in set-builder notation. Additionally, it is unnecessary to include the condition that $x$ is in the set of real numbers.
  • #1
samir
27
0
Hi!

I want to define a sine function that is discontinuous at multiples of $\pi$. The multiplier is to be an integer.

How can I do that?

I am thinking about something like this:

$$f(x)=\begin{cases}sin(x) & x \in \Bbb{R} \\ \text{undefined} & x=n \cdot \pi | n \in \Bbb{Z} \text{ and } x \in \Bbb{R}\end{cases}$$

Is this a valid statement?

Not only does it need to be discontinuous, but also undefined at these multiples. I believe it is the only way to have such a function be discontinuous at those multiples. Let me know if you know of another way.

I tried to graph this on my calculator but it didn't work. I think I need to define the condition for undefined in terms of x. Can I do that?

It seems like my calculator wants me to define $n$ first before I use it. But how do you define a variable that holds an element of the infinite set of integers? I'm not so sure. If I could set the data type to integers, that would get me half way there. Since it is a programmable calculator it should be possible. This is more of a product usage question, so I will leave that off for now. What I really want to know is if my definition of such function above is mathematically correct? I can deal with the calculator and graphing later.
 
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  • #2
A function can only be continuous or discontinuous at points where it is defined, because the definition of continuity/discontinuity involves the value of the function at that point. So your function need to be defined at multiples of $\pi$, but its value there should be different from the value of the sine function at those points (which is $0$). So you should amend your definition to say that $f(x) = \sin x$ when $x$ is not a multiple of $\pi$, and $f(x) = 1$ (say, or $17.629$, or anything at all other than $0$), when $x$ is a multiple of $\pi$.
 
  • #3
I would write
$$
f(x)=\begin{cases}\sin(x), & x \ne\pi n\text{ for any }n\in\mathbb{Z}\\ \text{undefined}, & \text{otherwise}\end{cases}
$$
The conditions in the two branches should be mutually exclusive. The conditions $x\in\Bbb R$ and $x=\pi n$ for some $n\in \Bbb Z$ are not mutually exclusive. Further, the vertical bar is only used in the set-builder notation, e.g., $\{x\in\Bbb R\mid x>0\}$ (and in this case it is written using the LaTeX command [m]\mid[/m] to create correct spacing on both sides). It does not have a universally accepted meaning of "such that" in other contexts. Finally, it is superfluous to write $x=n \cdot \pi | n \in \Bbb{Z} \text{ and } x \in \Bbb{R}$ because $x=\pi n$ and $n\in\Bbb Z$ imply $x\in\Bbb R$.
 

FAQ: Define a discontinuous sine function?

1. What is a discontinuous sine function?

A discontinuous sine function is a type of mathematical function that is defined piecewise, meaning it has different rules for different intervals or sections of its domain. This function is not continuous, meaning there are breaks or gaps in its graph.

2. How is a discontinuous sine function different from a regular sine function?

A regular sine function is continuous, meaning there are no breaks or gaps in its graph. It follows the same rule for all values of its domain. A discontinuous sine function, on the other hand, has different rules for different intervals of its domain, resulting in breaks or gaps in its graph.

3. What are the rules for a discontinuous sine function?

The rules for a discontinuous sine function vary depending on the interval of its domain. For example, in one interval, the function may follow the rule y = sin(x), while in another interval, it may follow the rule y = -sin(x). These different rules create breaks or gaps in the graph of the function.

4. What does a graph of a discontinuous sine function look like?

The graph of a discontinuous sine function will have gaps or breaks in it, where the function follows different rules for different intervals of its domain. These gaps or breaks are typically indicated by open circles or dashed lines on the graph.

5. What are some real-world applications of a discontinuous sine function?

Discontinuous sine functions can be used to model various real-world situations, such as the flow of electricity in a circuit, the motion of a pendulum, or the vibrations of a guitar string. These applications involve situations where the function follows different rules for different intervals, resulting in a discontinuous graph.

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