Hi again!
I thought I would finish off my previous posts on discontinuity by discussing asymptotic discontinuity. So let's focus on these alone in this thread.
I'm not familiar with the origin of the term "asymptote", but from what I can tell, it has asymptotic discontinuity to thank for for...
I earlier posted about point discontinuity. It became overwhelming pretty quickly. Now I would like to focus this thread at jump discontinuity specifically, if you don't mind me posting multiple threads about discontinuity.
From what I understand, "jump discontinuity" occurs where the left-hand...
So the point needs to be an element in the domain, and the limit at that point has to be equal to the function value at that point? I say point, but I really mean x value.
$$\lim_{{x}\to{a}}f(x) = f(a)$$
So unless we have a graph of the function, we have to study the the limit at any given...
Hi! :)
This is what I was getting at. I like both description, but I would probably prefer the second description. I think you know me by now! I like symbols! :D
This looks a lot like a piece-wise function?... is it? Piece-wise relation perhaps?
Given the following pattern.
$$(-1)^{0}=1$$
$$(-1)^{1}=-1$$
$$(-1)^{2}=1$$
$$(-1)^{3}=-1$$
$$(-1)^{4}=1$$
$$(-1)^{5}=-1$$
$$(-1)^{6}=1$$
$$(-1)^{7}=-1$$
$$\ldots$$
In words, we might say that the power of a negative number is:
Positive for even exponents.
Negative for odd exponents.
How...
$y$ intercept is at $x=0$.
$$x=0 \Leftrightarrow f(0)$$
$x$ intercept is at $y=0$.
$$y=0 \Leftrightarrow 0=\frac{x^3}{x^3+1}$$
$$f(0)=0$$
$$0=\frac{x^3}{x^3+1}$$
$$x=0$$
How do you find asymptotes?
This is a rational function. In other words the rule of the function is a rational...
Hi!
I'm looking at some piece-wise function right now and I can't help but wonder what all these parts are called. I'm learning to use and write this type of functions now and I think I have a pretty good understanding of how they work. I even took the extra step of learning some "set builder...
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:
$$f(x)=\begin{cases}sin(x) & x \in \Bbb{R} \\ \text{undefined} & x=n \cdot \pi | n \in \Bbb{Z} \text{ and } x \in...
Hi!
We have discussed complex numbers in class and their conjugates. From what I understand only the imaginary unit is conjugated. But I wonder if there are such things as real conjugates of complex numbers?
Given the following points:
$$A=(-2+i)$$
$$B=(2+3i)$$
$$C=(-4-3i)$$
$$D=(-4+i)$$
I...
Did you intentionally use "very close" and "close"? As to say that $f(x)$ gets very close to $L$ as $x$ gets only "close" to $a$. I would expect them to be equidistant. But that might depend on the function rule.
By "compare closeness" you mean the absolute distance between two numbers?
What...
Let me see if I understand this first part correctly.
$$f:\Bbb{R} \to \Bbb{R}$$
$$f(x)=2x$$
If $x=a$ is a number in the domain of the function, then we have the following limits.
$$\lim_{{x}\to{a^{-}}}f(x) = f(a)$$
$$\lim_{{x}\to{a^{+}}}f(x) = f(a)$$
Left limit is equal to the right limit...
Here is a simple linear function with one exclusion point.
$$f(x)=2x, x\neq 2$$
We can rewrite this as:
$$\displaystyle f(x)=\begin{cases}2x, & x \in \Bbb{R} \\ undefined, & x=2\end{cases}$$
As with the previous examples, this function is undefined at $x=2$. So for this reason the function...
At the very basic level, I understand this notion of discontinuity. But I am looking to understand this better, at a deeper level. Because I know I will need to know this well as I start to explore piece-wise functions.
Why do we bother drawing graphs? I want to be able to tell if a function...
Thank you guys! (Smile) I am really impressed by your display of knowledge here! (Clapping)
This is very helpful. I will surely revisit this thread often as I explore these topics.
Let me get back to some examples again. I will jump back to the linear function again.
$y=2x$
$y: X\to Y$...
Yes, I have seen this notation too. But this alone does not define the domain of the function $f$. It would need to be not only described as the "domain of f". It would also need to be defined using set builder notation.
Maybe something like this.
$\operatorname{dom}(f)=\left\{x\in \Bbb{R} | x...