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Can every possible curve/parabola shown on a graph (for example $x^2$ or somthing much more complicated) be expressed in an equation like $y=x^2$. Or are there some lines you can't express?

Vent
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    $;y=x^2;$ is not a line but a parabola. Perhaps the most general term is curve or function graph. – DonAntonio Mar 18 '16 at 14:19
  • Edited, thanks. – Vent Mar 18 '16 at 14:22
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    The number of mathematical expressions is countable, while the number of real-valued functions is not. – Andrew Dudzik Mar 18 '16 at 14:24
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    @Slade the number of mathematical expressions is not countable, think of infinite series. Or am I getting it wrong? – Henrique Augusto Souza Mar 18 '16 at 14:25
  • What restrictions are you placing on the curve? Presumably you want it to be continuous. But that is a weak restriction. You might want it to be smooth? – almagest Mar 18 '16 at 14:26
  • @Henrique I do not see the problem. – Andrew Dudzik Mar 18 '16 at 14:26
  • The answer is yes; trivially let $C$ be a curve. The question should rather be, can every curve be described as $F(x)=0$ where $F$ is a composition of a finite number of elementary functions (or, if you wish, plus a few more, like the Gamma function, etc). Then, the answer is no. – Pantelis Sopasakis Mar 18 '16 at 14:27
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    @Slade "The number of mathematical expressions is countable" That's an interesting statement. It directly leads to the question: "What is a mathematical expression?" – Friedrich Philipp Mar 18 '16 at 14:29
  • Not only can most curves not be written down. Most real numbers (or constant functions) cannot be written down. The numbers that can be written down fall under the heading of Computable Numbers, of which there are only countably many. It should be easy to convince yourself that if a number can be specified by a finite sentence of words and symbols then we can get a computer to spit out the decimal expansion for that number (however spitting out the entire expansion will take forever.) – Daron Mar 18 '16 at 14:47
  • @Slade: Of course if you can specify the elements of a sequence, for example $a_n = (-1)^k/(2k-1)$, you can specify the infinite sum (if it converges), for example $-\pi$. But they we have to ask how many sequences can we actually "write down". This has the same problem as for real numbers -- only countably many sequences are computable. Indeed looking back to real numbers shows there are only countably many computable sequences with entries in ${0,1,2,3,4,5,6,7,8,9}$ – Daron Mar 18 '16 at 14:51

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