Question

give an example of function a continuous but not bounded on (a,b)​

Answer 1

Step-by-step explanation:

The ceiling function and the floor function possess jump discontinuities at x=1 and x=2 respectively. If you want to exclude the endpoints of the interval, you can always shift the function by translation.

However, if you want a nowhere continuous function, consider the Dirichlet function denoted by D(x)

D(x)={1, if x is rational0, if x is rational.