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If I draw any kind of shape on a piece of paper (possible discontinuous), would it be theoretically possible to represent the shape in a mathematical function?

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  • $\begingroup$ Sure as long as you're ok with it being parametric $\endgroup$
    – Alexander Gruber
    Apr 6, 2018 at 17:58
  • $\begingroup$ Any shape you can draw on a paper you can also describe in $x,y$ coordinates. Then you code color into $f(x,y)$. $\endgroup$ Apr 6, 2018 at 17:58
  • $\begingroup$ If you mean a function $y = f(x).$ Every $x$ value will have at most one $y$ value. And the graph of this function must pass the vertical line test. But, if you want a parametric curve. $(x,y) = (f(t), g(t))$ Then yes, it can be done that way. $\endgroup$
    – Doug M
    Apr 6, 2018 at 18:01
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    $\begingroup$ Theoretically, yes, because at every time $t$, your pen is at some point $(x(t),y(t))$ on the paper. But most likely you cant represent the functions $x(t),y(t)$ using elementary functions $\endgroup$ Apr 6, 2018 at 18:01
  • $\begingroup$ Oh, yes! I forgot! Should I edit and change the question? -Doug M $\endgroup$ Apr 6, 2018 at 18:06

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You have to be careful here. By definition, a function has one possible output for any given input. So if you want your function defined as some $y = f(x)$, then no not every shape can be written as a function. Any shape that has two points directly above each other (relative to the x-axis) cannot be written as a function, even a piecewise one.

However, if you are defining "a function" as any parametric curve $(x,y) = (f(s),g(s))$, then yes this can be done for any shape that can be drawn without lifting your pencil off of the paper. Simply define your parameter $s$ to run along the same path your pencil moves and your $f$ and $g$ are therefore well-defined too.

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  • $\begingroup$ And if I lift the pencil? I mean if the shape is not continuous everywhere? $\endgroup$ Apr 6, 2018 at 18:09
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    $\begingroup$ @ShuvoSarker That would be valid too technically. But you would have to describe the shape as a piecewise parametric curve. So for each continuous curve, you write $(x_i,y_i) = (f_i(s),g_i(s))$, and you have $i = 1,...,n$ with $n$ distinct curves. $\endgroup$ Apr 6, 2018 at 18:11
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As a function? No because no function can have more than one output value. This means that if your draw a line parallel to the y-axis, the line should cut the function at one and only one point.

But you will always find a relation that gives the image. And there's always an equation for every picture if you don't constrain your choice of functions, you can use, too much.

For example, there's no function whose graph is a circle but the equation $$x^2+y^2=1$$ gives a circle of radius 1.

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