Proving Product of Regulated Functions is Regulated

[C.E]

1 Show that the product of two regulated functions is regulated.



2. A function is regulated if it is the limit of a sequence of step functions.



3. I let f,g be regulated and let a_n, b_n tend to f, g respectivley. I can show that for any x, a_n (x) . b_n (x) tends to f(x).g(x) (i.e. pointwise convergence). Is this sufficient or do I need to show uniform convergence? If so how do I go about it?

 

[ystael]

You do need to show uniform convergence, since to show $$fg$$ is a regulated function, you need to exhibit a sequence of step functions converging uniformly to $$fg$$.

I think one needs some additional information; is the domain a compact closed interval? The function $$f(x) = x$$, for instance, is not a uniform limit of step functions when you take the domain to be all of $$\mathbb{R}$$.

Assuming the domain is bounded, think about $$\sup_x |f(x)|$$ or $$\sup_x |g(x)|$$.

 

[C.E]

Sorry I probably should have said, the two functions f,g are on the closed real interval [a,b] for some real a,b.