# Can a function be only defined graphically ??

are there functions that can be only defined graphycally ?

for example the soultion of an integral equation

$$f(s)= \int_{0}^{\infty}dxK(s,x)f(x)$$

if i find numerically a graph of the function $f(x)$ on the interval $(-1,1)$ does it mean that the function $f(x)$ exists on this interval ??

for example let be teh function defined implicitly by

$$f^{-1}(x)=x+cos(x)+cos(sin(x))$$

i can use MATHEMATICA to graph it since http://www.wolframalpha.com/input/?i=inv%28x%2Bcos%28x%29%2Bcos%28sin%28x%29%29

so if teh graphic of the function exists then the function itself exists ?=

of course i am a physicst so for us if we can draw a graphic of the problem then we considered it solved.

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Define function, define its graph –  Ilya Jan 24 '13 at 12:01
The functions you ask about are not "only defined graphically". The first one is defined to be the solution of a certain integral equation. The second is defined to be the inverse of some explicitly given function. Those are (for a mathematician) definitions! –  Gerry Myerson Jan 24 '13 at 12:02
ok tahnks.. you are right :) –  Jose Garcia Jan 24 '13 at 12:16

I'm not sure about the first sentence. The dilogarithm, $\sum_1^{\infty}z^n/n^2$, is defined explicitly and is a series but is not generally considered an elementary function. There are many examples of this phenomenon. Also, I see nothing in the original question that makes me think the Axiom of Choice is involved. –  Gerry Myerson Jan 24 '13 at 22:51
@GerryMyerson Regarding the "explicit definition" I didn't mean to state that series and infinite products are to be considered elementary functions. I guess that an infinite series is "quite explicit" since it can be written somehow as $f(x)=\ldots$. What I was talking about were such "explicit functions", which include elementary functions. –  AndreasT Jan 24 '13 at 23:16