Uniformly Continuous and locally Lipschitz but not Globally Lipschitz Function on a "Connected but not compact" set

I know such a function exists but I can’t find an example. I have the famous example f:]0,inf[ --->R f(x)=sqrt(x) function. But I can't find any other function which is Uniformly Continuous and locally Lipschitz but not Globally Lipschitz (on a connected but not compact set) Can someone help me please?