Another regulated functions question

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The discussion centers on the mathematical concept of weakly increasing functions and their approximation through step functions. Specifically, it addresses the requirement to demonstrate that a weakly increasing function f: [a, b] → R can be represented as the limit of a sequence of step functions. Participants emphasize the importance of visualizing the function's graph, suggesting that the approximation should mimic how a computer monitor displays images, using square pixels to represent the function's values on a grid.

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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.
 
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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 :)
 
Last edited:
Surely different choices of f would look different wouldn't they? How do this help?
 
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.
 
Sorry, but I don't know how a computer monitor would approximate the graph of f, can you please elaborate?
 
Think of square pixels -- approximate the graph of f by something you might draw using the edges of a square grid.
 

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