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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
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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.

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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
    
@GerryMyerson Concerning the axiom of choice, since the asker talks about computer-plotted graph I thought about machine precision. The value of the function in each point is not exact, rather is known to lie in a certain set determined by the graph's precision. In this sense the axiom of choice ensures that there exists a function whose 'approximated' graph is the output of the program. I agree that this cannot be a definition since it lacks uniqueness, but it was the closest guess I came up with regarding a graphical approach. –  AndreasT Jan 24 '13 at 23:30
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