Proof Lebesgue-stieltjes eqals Riemann-Stieltjes integral in this case?

I am looking for a reference or a direct answer for a proof of this statement:

Assume you have a random variable X. Define F(x) to be $P(X\le x)$,this function is bounded, monotone, non-deacreasing and right-countinuous and $\lim_{x \rightarrow -\infty} =0$. Define $\mu$ to be theLebesgue-Stieltjes measure of the distribution function $F$. Assumethat $-\infty < a \le b <\infty $. Assume that $f$ is a continuousreal-valued function. Define $\int_a^b f dF(x)$ to be theRiemann-Stieltjes integral.

We then have

$$\int_a^b f dF(x) = \int_{(a,b]}f d\mu.$$

According to the answers here:

Is expectation Riemann-/Lebesgue–Stieltjes integral?

,this statement seems to be true.

I am able to show this when $f$ is the constant function, but not generally when it is continuous.

PS: I am not sure if I should have written $[a,b]$ instead of $(a,b]$? This is important for the case have $P(X=c)>0$ for a real number c. This also makes me unsure if I should have $a<b$ or if I can have $a \le b$?