Let X be a open subset on Rn.
Let S be the set of all real, bounded, and continuously diferentiable functions on X, and rho is the sup norm.
Is (S,rho) a complete metric space?
Could you give me a counter-example if it is not?
But, let's assume you meant to take the open subset of to be and the function . Is that even differentiable at ?
I haven't checked it out, but what about with . This is differentiable and continuous, and I'm pretty sure Cauchy but it "wants" to converge to which is only differentiable at , not continuously differentiable.