Monotonocity of Integrals, if and only if? Assume that $f(x)$ and $g(x)$ are Riemann Integrable and that $\int_a^W f(x) \, dx<\int_a^W g(x) \, dx$ for all $W \in [a,b]$.
Does it follow that
$$ f(x) < g(x)  $$
for all $W \in [a,b]$.
Is it true with weak inequalities?

I think this doesn't hold in general. Fix some $c \in [a,b]$ and take for example $g : [a,b] \to \mathbb{R}$, with $g(x)=1$, and
$$
f : [a,b] \to \mathbb{R}, \quad x \mapsto 
\begin{cases}
0 & x \in [a,b]\setminus \{c\}, \\
2 & x=c.
\end{cases}
$$
Then $f$ and $g$ are Riemann integrable on $[a,b]$, because $g$ is continuous and $f$ is bounded and has only one discontinuity. We indeed have that
$$
\int_a^W f(x) \, dx < \int_a^W g(x)\, dx
$$
for all $W \in [a,b]$. However, for $x=c$, we have $f(c)=2 > 1 = g(c)$.
