Let's say that I have a function $y = f_1(x)$. I have two other functions, $y = f_2(x)$ and $y = f_3(x)$, that I want to compare with the first function. Specifically, I want to see which one of $f_2$ and $f_3$ can best $approximate$ the function $f_1$. How do I go about doing this? All three functions are continuous.
I tried to think of an approach based on "error" that we see in regression analysis. For discrete data, we can calculate all kinds of errors that evaluate goodness of fits. For instance, if $f_1$ were a discrete function with values $\{y_1, y_2, \cdots , y_N\}$ corresponding to $\{x_1, x_2, \cdots, x_N\}$, the Mean Square Error (MSE) of $f_2$ would be given by $$ \text{MSE}_2 = \frac{1}{N} \sum_{i = 1}^N |y_i - f_2(x_i)|^2$$ and ditto for $f_1$. My question is, can we extend this to the case where $f_1$ is continuous? My initial thought was yes, since we can simply generalize the sum to an integral, and consider infinite data-points, meaning $N\to \infty$. This way, defining $\delta x \equiv 1/N$, I can argue $\delta x \to dx$ as $N \to \infty$. The new MSE would then be $$\text{MSE}_2 = \int_{a}^b |f_1(x) - f_2(x)|^2 \, dx.$$ I can then do the same for $f_3$ and compare the two results: whichever function gives me a lower MSE is a better approximate to $f_1$. Is this sound? Moreover, is this a good way to find out how well one function approximates the other, as is my initial goal?
There are as many ways to measure how close two continuous functions (defined, say, on a closed and bounded interval $[a,b]$), as there are metrics on the set of those continuous functions.
The (square root of the) MSE$_2$ you are using is one metric on continuous functions. Another is the uniform metric, given by
$$d(f_1,f_2)=\max_{a\leq x\leq b}|f_1(x)-f_2(x)|$$
It is clear that MSE$_2$ is at most $(b-a)d(f_1,f_2)^2$, but two continuous functions can be close in the MSE$_2$ sense, but not so close in the uniform sense.