Another regulated functions question

[C.E]

1. suppose f:[a,b]->R is a weakly increasing function, i.e. u<v gives f(u)<=f(v). Show that f is the limit of a sequence of step functions.

3. I have no idea how to start this. Please help.

 

[ystael]

If you were to draw the graph of $$f$$ on a computer monitor, what would the picture look like?

Now buy successively better computer monitors :)

 

[C.E]

Surely different choices of f would look different wouldn't they? How do this help?

 

[ystael]

Of course the sequence of approximating step functions will depend on the particular function $$f$$. My hint is that you should approximate the graph of $$f$$ roughly as a computer monitor does.

 

[C.E]

Sorry, but I don't know how a computer monitor would approximate the graph of f, can you please elaborate?

 

[ystael]

Think of square pixels -- approximate the graph of $$f$$ by something you might draw using the edges of a square grid.