Math Help - Compare two functions

I have two functions that I want to compare - call them f(x) and g(x), x is real, and f and g are R -> R. What is the most appropriate way to compare these functions? I should note that both of these functions are Riemann integrable - they are bounded and continuous.

One comparison could be h(x) = f(x)-g(x). Taking this a step further, we could integrate h to produce a numerical value for the difference between these two functions.


Another comparison is h(x) = f(x)/g(x). This function won't be continuous or bounded at the roots of g. We could also do log(h(x)).


Are there other ways to compare functions?

Hey heaviside.

What specific thing do you want to compare? If you say "closeness", then you need to say what "closeness" is.

A typical way to compare things is to use norms and a typical norm that is useful is the 2-norm where ||f,g|| = (f(x)-g(x))^2 where the norm is Integral over some region ||f,g||dx.

Thanks chiro

Yes - when I wrote this, I was thinking closeness. Or similarly, identifying how different two functions are when these functions are very similar but only slightly different. I know this is a lot of handwaving since I have not specified what 'similarity' is nor 'slightly different'.

I like the 2-norm approach. I've seen it many times but didn't think about using it in this instance. However, I can see that it is a better measure than simply f-g.