How can I deduce the following equation ?:
$$ \lim_{m \to \infty}\prod_{i = 0}^{{\large tm - 1}} \left[\,1 + \frac{r\left(\,i\,\right)}{m}\,\right] = \exp\left(\,\int_{0}^{t}r\left(\,s\,\right)\,\mathrm{d}s\,\right) $$
where $t > 0$ and $r\left(\,i\,\right)$ a real valued function.
I can figure out the series expansion of $\mathrm{e}$, and it makes somehow sense as you can think of $r$ as the average "rate of interest" for example, but what exactly is the math behind it ?.
HINT:
There is a suspected typo in the OP. The argument of $r$ should be $i/m$, not $i$ alone.
Then, we can write
$$\log\left(\prod_{i=0}^{\lfloor tm-1\rfloor} \left(1+\frac{r(i/m)}{m}\right)\right)=\sum_{i=0}^{\lfloor tm-1\rfloor} \log\left(1+\frac{r(i/m)}{m}\right)$$
Now use the fact that $\log\left(1+\frac{r(i/m)}{m}\right)=\frac{r(i/m)}{m}+O\left(\frac{r(i/m)}{m}\right)^2$ as $m\to \infty$.
Finish by evaluating the resulting Riemann Sum.