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So given that a taylor-series expansion is exact for an infinitely-times differentiable function, how does this constraint permit a bijective mapping of the function to a countable list of coefficients? How would you prove this is ok?

Additionally There exists several constraints that permit such a mapping, another example would be the Fourier coefficients for any periodic function. So to generalize the question is there a way to define the amount of information a function holds based off the constraints given?

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    I don't understand your first sentence. – 2017-01-19
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    How is it that a continuous function from $x=a$ to $x=b$ not contain all points $x\in(a,b)$ and likewise all values $f(a)\le f(x)\le f(b)$ for $x\in(a,b)$ by the intermediate value theorem? – 2017-01-19
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    "taylor-series expansion is exact for an infinitely-times differentiable function" That (if I understand what you mean) is not true: http://math.stackexchange.com/questions/924457/does-the-taylor-series-of-an-infinitely-differentiable-function-converge-and-if – 2017-01-23

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