Product Rule and Reflection for Indefinite Riemann Integrals

Let $a<b$ be real numbers. All integrals below are Riemann integrals of continuous functions — each interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric — which exist by claim 3 of the integral toolkit on a compact interval, with the degenerate-interval convention of Mean-Square Riemann Integral of a Family of Random Variables.

1. (Product rule) Let $f,g:[a,b]\to\mathbb{R}$ be continuous, let $c_F,c_G$ be real numbers, and define

Then $F$, $G$, and $Fg+fG$ are continuous on $[a,b]$, and for every $t\in[a,b]$

2. (Reflection) Let $\varphi:[a,b]\to\mathbb{R}$ be continuous. Then the function $\rho\mapsto\varphi(a+b-\rho)$ is continuous on $[a,b]$, and for every $t\in[a,b]$