Can a function be only defined graphically ??

Question

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.

Answer 1

The functions you can define explicitly are the so-called Elementary Functions (including floor, ceiling, series, infinite products and others). Note that functions need not be elementary in order to be well defined. On the other hand, your question about a graphical definition may be linked to the Axiom of Choice: if you have a collections of non-empty sets and you pick ('choose') one element from each, then the collection of the elements you chose defines a new set. The functions you are questioning about probably are related to choice functions.