computability and uncomputability

Question

1) Suppose $f$ is an increasing function from $\mathbb N \to \mathbb N$ $(i.e., if x\ge y, then \space f(x) \ge f(y)).$ Is there necessarily a program which computes $f$?

2) Suppose $f$ is a decreasing function from $\mathbb N \to \mathbb N$ $(i.e., if x\ge y, then \space f(x) \le f(y)).$ Is there necessarily a program which compute $f$?

Can anyone help me out with this problem. I have no idea how to do this.

Answer 1

1. Just give an example of a non computable increasing function to prove it wrong.
2. If $f$ is decreasing, consider $L=\lim_{x\rightarrow\infty}f(x)$. Then $f(x)=L$ for all $x$ except a finite number of $x$. Then you should be able to make a program that computes $f$.