Another really basic question this time regarding integration.

[Mathguy15]

Given a function f define a new function Sf(x) by summing up all values of f(hj)
where 0 ≤ jh < x. That is, if k is such that kh is the largest below x, then
Sf(x) = h[ f(0) + f(h) + f(2h) + ... + f(kh) ]
We call Sf also the ”integral” or ”antiderivative” of f.

The teacher who wrote the lecture notes I'm reading through gives an example of integration. He evaluates Sf(x) for f(x)=1. I don't understand the first sentence:

We have Sf(x) = 0 for x ≤ h.

Why? Sorry for being such a n00b, but I don't understand. Please help me.

Thanks,
Mathguy

By the way, he verifies that the js in the definition are integers.

 

[pwsnafu]

We call Sf also the ”integral” or ”antiderivative” of f.

Err, the integral is what you get if you take the limit h -> 0.
Is that word-for-word what is written there?

We have Sf(x) = 0 for x ≤ h.

That doesn't seem right. jh is allowed to equal 0, so the largest integer k such that
0 ≤ kh < x ≤ h is when k=0. So Sf(x) = h f(0) = h.

Edit: Maybe he means x < 0?

 

[Mathguy15]

Well, Yes, that is word-for-word, but I think he's doing a "preliminary" definition before the real definition. And I was thinking the same thing, because Sf(x) isn't defined for x<0.