In theory, under B-S framework the delta is smooth, so the delta can't jump.
But the delta and gamma can be very large under some situation.

Correct, the "jump" we are talking about here is slope equal or close to infinity, not continuous or discontinuous. Maybe it is a little misleading. Sorry about that :-)

I work in the industry, what I said is the common practice they use in practice.

We are not talking about some details here. I know in Pual's book Chapter 7 it talks about how to estimate greeks more efficient and reduce the estimation error. There are of course many ways to make the estimation more precise and efficient what I said is just a general answer.

Just like Euler scheme is still a good way to simulate path although there are many other improvements like including second order errors such as Milstein method etc. But you can not say Euler scheme is wrong, although it is a naive way. :-)

好赞啊

use the simple way to calculate delta.and according to the book delta=dC/dS0,so y P1-P0 not
P1-P0/S1-S0. how exactly to static hedging for hedging barrier option.（Up and out）.monitoring delta daily and change position dynamically？ thx

国内障碍期权一般是指数为标的，股票的不做

之前的外文回答特别好，面对jump，某些跳跃值需要特殊处理，在这种情况下，barrier call put的delta有其有效区间。