Topology/ Metric on possibly unbounded functions I am trying to think of a topology (possibly metric, as I am more used to think about things in metric spaces) on possibly unbounded functions (on $\mathbb{R}$) such that 
1) convergence in that defined topology implies pointwise convergence and 
2) the limit of continuous functions is continuous itself. 
I know that a sequence of functions converges to a function $f$ under the metric derived from the uniform norm if and only if converges to $f$ uniformly. And I know the existence of the Uniform limit theorem. (link ) Also convergence in uniform norm implies pointwise convergence.
The problem is that the sup norm/ uniform norm is defined on bounded functions, whereas, I want to think about general functions. 
Could I use a metric, $d'=\dfrac{d}{1+d}$ where d is the uniform metric, and 1 when d is infinity? The "normalization" makes sure that I do not get infinite values while getting the distance between functions. And it seems to be that given how this new metric is defined, it would satisfy 1) and 2). Do you guys think I am on the right track?
Also, is there a more obvious metric that I am missing?
 A: The topology you are looking for is, I think, the topology of uniform convergence on compact sets. It is the topology defined on $\mathcal{C} (\mathbb{R}, \mathbb{R})$ such that:
$$f_n \to_{n \to + \infty} f \ \Leftrightarrow \ \lim_{n \to + \infty} \max \{|f_n (x) - f(x)| \ : \ x \in K\} \ \ \forall \text{ compact } K.$$
It is actually sufficient to check convergence on an exhaustive family of compact sets. On $\mathbb{R}$, take for instance $K_M := [-M, M]$. Hence:
$$f_n \to_{n \to + \infty} f \ \Leftrightarrow \ \lim_{n \to + \infty} \max \{|f_n (x) - f(x)| \ : \ x \in K_M \} \ \ \forall M \geq 0.$$
Finally, if you want a single metric, put $\|f\|_{\infty, K} := \max \{|f_n (x) - f(x)| \ : \ x \in K \}$. Then a possible metric is:
$$d(f,g) := \sum_{M \geq 0}2^{-M} \frac{\|f-g\|_{\infty, K_M}}{1+\|f-g\|_{\infty, K_M}},$$
although in all examples I have seen, it has been more convenient to work with the definition (first equation) than with this explicit metric.
This construction works for functions on more general spaces (we need only $\sigma$-compactness, so any open set or any manifold is fair game), for higher regularity...
A: Here is a metric which describes uniform convergence on $\mathbb R$: $D(f,g)=\sup\lbrace \min\lbrace|f(x)-g(x)|,1\rbrace: x\in\mathbb R\rbrace$.
