Defining a function in neighborhoods of points

I want to define a function g(s) such that g(s_):=f(s) in a (arbitrarily small open) neighborhood around s1 and g(s_):=0 in a (arbitrarily small open) neighborhood around s2.

How do I do that? It looks like Piecewise requires actual numbers to be inputted, but I don’t have numerical values of s1 and s2, so I don’t think I can use that.

The goal would be to get it so that

g(s1)==f(s1)
g'(s1)==f'(s1)
g(s2)==0
g'(s2)==0