Find a real function knowing its values for all natural numbers

Question

Consider a function $f(x) : \mathbb{R} \rightarrow \mathbb{R}$ smooth enough such that $f(\mathbb{N}) \subseteq \mathbb{N}$.

Is there some methodologies to find another function $g(x): \mathbb{R} \rightarrow \mathbb{R}$ smooth enough such that:

$$f(n) = g(n) ~\forall n \in \mathbb{N} \setminus \{n_0\}$$

and

$$f(n_0) \neq g(n_0) \in \mathbb{N}?$$

In general, given a sequence $\{a_n\}\subset \mathbb{N}$, can I build a real smooth function such that $f(n) = a_n ~\forall n \in \mathbb{N}$?

Additions

"Smooth enough" is misleading. Sorry for this. I mean that the function must not be piecewise and it must be in $\mathcal{C}^{\infty}$.

In practice, I'm facing with the following problem:

Suppose that $f(x) = x$. I'd like to derive a function $g(x)$, such that $n_0 = 1$ and $g(n_0) = 3$ and $f(x) = g(x)$ for all $x \in \mathbb{N} \setminus \{1\}$.

Answer 1

Yes, you can use a linear interpolation to do that.

http://en.wikipedia.org/wiki/Interpolation#Linear_interpolation

Answer 2

An equivalent question, if I understand this correctly, would be if we can find a smooth function h=f-g so that h(n)=0 on all naturals n except one.

I think we can do this with an appropriately scaled bump function.

http://en.wikipedia.org/wiki/Bump_function

Take h=Ψ as defined on the wiki page, and then let g=f+h.

Answer 3

The function $${\rm sinc}(\pi x):=\cases{{\sin(\pi x)\over\pi x}\quad&$(x\ne 0)$\cr 1&$(x=0)$\cr}$$ is an entire function which is $=1$ at $0$ and $=0$ at all integers $\ne0$.