Is it possible to find the $n$th digit of $\pi$ (in base $10$)? Is it possible that there exists some function $f:\mathbb N_1\to \{0,1,2,3,4,5,6,7,8,9\}$, where $$f(1)=\color{red}1, f(2)=\color{red}4, f(3)=\color{red}1, f(4)=\color{red}5, f(5)=\color{red}9, f(6)=\color{red}2, \ldots \quad ?$$
I know that such a function, if it exists, cannot be a polynomial, since all polynomials are finite-dimensional, whereas the sequence containing the digits of $\pi$ has infinitely many elements, but could it be that there is some function (even if we cannot explicitly state it) which gives the $n$th  term of the sequence containing the digits of $\pi$?
If this is true, and such a function does exist, can the same be said for all transcendental numbers?

Edit: to be absolutely clear, I am looking for the explicit function $f$ such that $f(n)$ is the $n$th digit of $\pi$ (in base $10$).

What exactly means explicit? For a given $n$, there is certainly an algorithm that computes the $n$'th digit of $\pi$. 
But the answer is no if you ask about some other real numbers. There are non-computable numbers that encode the halting problem, for example. There is no chance to have an algorithm computing the digits of some numbers in general -- and hence not any reasonable "formula" composed of common basic operations for those numbers.
