Is continuous compounding as used in math textbooks actually relevant to the real world?

I've always struggled when teaching this, mainly because of relevance. The idea is that if 12 percent interest is calculated at 6 percent twice a year or 1 percent every month and on to the limit you get a higher effective interest rate. But who cares? If a bank is advertising 12 percent yearly interest that does actually mean you get 12 percent, and in one month you'd get the 12th root of 1.12 right? Same with credit cards? So where exactly does this weird e^rt thing actually come in any scenario where people need to know actual exponential growth rates? For population growth 10 percent growth means it grew 10 percent in a year, not some theoretical upper limit of continuous compounding?

Edit: I don't think I explained this well. I'm not talking about the concept of exponential functions being continuous. That can be achieved by 1.12^t = e^((ln1.12)x) if your really want e in there. I'm talking specifically about writing that as e^(.12t), which ends up in a yearly rate higher than 12 percent.