Frigyes Riesz originally proved the Riesz Representation Theorem on $ C[0,1] $ -- here is his 1909 paper in English (original French). He builds a real valued function $ \text{A} $ on $ [0,1] $ satisfying
$$ {{\LARGE{\sum_i}} {{\left|\text{A}(x_i)-2\text{A}\left({{x_i+x_{i+1}}\over 2}\right)+\text{A}(x_{i+1})\right|} \over {x_{i+1}-x_i}}} \leq {M_{\mathcal{A}} \over 2} $$
for any partition $ \{0=x_0, x_1, x_2, \cdots x_{n-1} = 1\} $ and a fixed constant $ M_{\mathcal{A}} $, then says, "It follows that the derived numbers of the function $ \text{A}(x) $ exist and that these derivatives are functions of bounded variation."
My problem: By "derived numbers" he means the Dini derivative (upper right, say). Granted the expression in the sum is close to a difference quotient, I'm still not seeing why the upper right Dini derivative exists and is of bounded variation.
Any insight would be greatly appreciated.
Update: Riesz's 1909 paper cited above announced the Riesz Representation Theorem in a brief note. He gave more details in a followup paper in 1911. There he rewites the sum above as:
$${{\LARGE{\sum_{\large{k=1}}^{\large{m-1}}}} \left|{{\text{A}(x_{k+1}) - \text{A}(x_k)} \over {x_{k+1}-x_k}} - {{\text{A}(x_k) - \text{A}(x_{k-1})} \over {x_k-x_{k-1}}} \right|} \leq G,$$
where $x_k$ is the midpoint. Putting $D_k = \large{{\text{A}(x_k)-\text{A}(x_{k-1})} \over {x_k-x_{k-1}}}$, this reads:
$$\sum_{k=1}^{m-1} |D_{k+1} - D_k| \leq G,$$
which makes it clear that the difference quotients are of bounded variation, hence bounded, so $\text{A}$ has finite upper Dini derivatives.
I wrote up the recapituation here: Riesz Proves the Riesz Representation Theorem. His proof was highly concrete, based on simple piecewise linear functions on $[0,1]$ — easy to understand and a real work of art.