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## Main Question or Discussion Point

I'm having trouble visualizing the riemann-stieltjies integral...

Our textbook states:

We assume throughout this section that F is an increasing function on a closed interval [a,b]. To avoid trivialities we assume F(a)<F(b). All left-hand and right-hand limits exist...We use the notation

[tex]F(t^-)= lim_{x \rightarrow t^-} F(x)[/tex] and [tex]F(t^+)= lim_{x \rightarrow t^+} F(x)[/tex]

For a bounded function f on [a,b] and a partition [tex]P={a=t_0 < t_1 < ... < t_n = b}[/tex] of [a,b], we write

[tex]J_F(f,P) = \sum_{k=0}^n f(t_k) [F(t_k^+) - F(t_k^-)][/tex]

The upper Darboux-Stieltjes sum is

[tex]U_F(f,P) = J_F(f,P) + \sum_{k=1}^n max(f, (t_{k-1}, t_k) [F(t_k^+) - F(t_{k-1}^-)][/tex]

I'm having trouble visualizing this...also, by F(x), do they mean the integral of f(x)?

Thanks in advance

Our textbook states:

We assume throughout this section that F is an increasing function on a closed interval [a,b]. To avoid trivialities we assume F(a)<F(b). All left-hand and right-hand limits exist...We use the notation

[tex]F(t^-)= lim_{x \rightarrow t^-} F(x)[/tex] and [tex]F(t^+)= lim_{x \rightarrow t^+} F(x)[/tex]

For a bounded function f on [a,b] and a partition [tex]P={a=t_0 < t_1 < ... < t_n = b}[/tex] of [a,b], we write

[tex]J_F(f,P) = \sum_{k=0}^n f(t_k) [F(t_k^+) - F(t_k^-)][/tex]

The upper Darboux-Stieltjes sum is

[tex]U_F(f,P) = J_F(f,P) + \sum_{k=1}^n max(f, (t_{k-1}, t_k) [F(t_k^+) - F(t_{k-1}^-)][/tex]

I'm having trouble visualizing this...also, by F(x), do they mean the integral of f(x)?

Thanks in advance