(From Morris Kline's Calculus)
Talking about continuous compounded interest, he says:
"The most important point about the above discussion is that continuous compounding of interest at the rate of 0.04 per year is an example of the phenomenon in which a quantity changes continuously at a rate that is applied constantly to the amount present at any instant."
The bolded part is very confusing to me. It sounds to me as if he's saying that at any instant during that year, the quantity grows at a rate of $\frac{0.04}{year}$, and I can't make sense out of that.
I understand $e$ defined as $\lim_{t \to \infty} (1 + \frac{1}{t})^t$. What I'm thinking is that if you broke down that $4\%$ into $4$ times $1\%$ compounded every 3 months, you wouldn't say at the second instance of $1\%$ return after 6 months that your fund is growing at a rate of $\frac{0.04}{year}$, or would you?
This is all very confusing...I'm really looking for intuition, whether geometric, physical or other. Thank you.