Solve for $f: \mathbb{R}\to\mathbb{R}\ \ \ $ s.t. 

$$f(n)=e^n \ \ \forall n\in\mathbb{N}$$
$$f^{(y)}(x)>0 \ \forall y\in\mathbb{N^\*} \ \forall x\in\mathbb R$$

Could you please prove that there exists an unique solution: $f(x)=e^x$?

(Anyway, this problem is not about fractional calculus)

$\mathbb N^\*=\{1,2,3...\}, \ \mathbb N=\{0,1,2....\}$

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How about try to construct a few functional spaces that intersect at one point?

Try Sard Theorem and Pre image Theorem.