Every function passes the vertical line test, but is everything that passes the vertical line test a definable function?

Every function passes the vertical line test, but is everything that passes the vertical line test a definable function?

As in, no matter what I draw on a graph, there exists in C^n an equation that results in the output of that graph?

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** if it passes the VLT

The question is how do you come up with a function that isn't defined by some equation? I mean, what other way is there to make up a function graph?

A function could be described point by point. It does not have to be one equation or even a finite number of equations.

Let [math] C [/math] be a curve in [math] R^2 [/math] that passes the vertical line test. Then for every [math] x_0 \in R [/math] there exists at most one ordered pair [math] (x_0, y_0) [/math] contained in [math] C [/math]. Then for every such pair we may define [math] C=f(x) [/math] such that [math] f(x_0)=y_0 [/math] and else is undefined.

You speak funny xD

The VLT is just a geometric way of verifying that you don't have one x-value related to two distinct y-values.

Assuming your graph satisfies the VLT, there will be some function that corresponds to it. However, it's not always (in fact, almost never) possible to define it with some equation of x and y values.

/thread
A function maps a set to another set. Anything you could possibly draw on the graph that passes the vertical line test could technically be expressed as a function by using your drawing as the assignment rule between the sets.

is there a vertical plane test for (x,y,z) graphs? do yoy have to do it 2 ways?

An "(x, y, z) graph" could either be a function from [math]\mathbb{R}^2[/math] to [math]\mathbb{R}[/math] or the other way around. For the first, a "vertical (perpendicular to the x-axis) plane test" would work, for the second, a "vertical line" test would be enough.

>However, it's not always (in fact, almost never) possible to define it with some equation of x and y values.
You obviously haven't done Fourier analysis. You may not be able to define it with simple functions but there will always be an equation that corresponds to your graph.

You're forgetting about non-continuous functions :^).

We can still use equations to describe those as long as we are strict about the domain.

If you are going to enumerate the domain you might as well enumerate the range

Yes because it can always be decomposed into infinitely many piecewise functions

>define function point by point exactly like x^3
>derivative is 0 at every x value

Is that how this would work out, since every f(x) is a constant, even though tgere is change between the two points?

Wew

I think a good definition for an equation we can agree on is that it's a string over a countable alphabet. Thus an equation has countable length. The only way to write an equation for a function [math]f : \mathbb{R} \rightarrow \mathbb{R} [/math] that to each number assigns a random real number, would be to enumerate all [math](f, f(x))[/math] pairs. But there are [math]\mathfrak{C}[/math] of them - an uncountable amount.

*[math](x, f(x)[/math] pairs

>I think a good definition for an equation we can agree on is that it's a string over a countable alphabet.
I disagree. See I spaced out before and couldn't think of the right word but piecewise is it.

If we have a graph of a horizontal line that alternates between two y values every point. Would Veeky Forums let me post a frog?

Functions are different than equations. the may be defined with equations but not always.

>vertical line test
American "education" never ceases to impress me

>You obviously haven't done Fourier analysis.
but have you done it, brainlet ?

>le piecewise "functions" are actually functions meme

oh please

Not a single person in your thread has answered your question.

The VLT applies to SUBSETS OF THE PLANE, also known as relations from reals to reals.

A relation is a *function* if there is exactly one element in the codomain (output) that is related to every element in the domain (input). The VLT just tells you that there is *at most one* output for each input, but rigorously speaking an output must exist for every input, otherwise it's a partial function. But most people don't care much about the difference and will just say it's "undefined" at certain places. So there you go.

But piecewise functions are functions.

In the abstract mathematical sense, functions are simply a set of ordered pairs.

The equation y=x does not look like a line to a mathematician. It looks like {(0,0),(25,25),(-69,-69),(2,2),..}

But it just so happens that we can generate that function with the equation y=x in the context of analytic geometry.

That's it. Some functions are harder to generate and that is where piecewise descriptions come in.

>You obviously haven't done Fourier analysis
And clearly, you didn't understand it.

When you write a function, like a square wave, as a series of sines and cosines, you're making a very precise meaning behind the "=" sign. That is, it's really a shorthand saying that the function and its Fourier representation are equivalent only in the [math]L^2 [/math] sense, with respect to your chosen Lebesgue measure. This is exactly the source of the Gibbs phenomenon.

See e.g. en.wikipedia.org/wiki/Convergence_of_Fourier_series and remember that the handwavy shit you learned in your one semester "math for engineers" course is just that: handwavy shit.

(You)