Talk:Regulated function

Banach space

I have changed the requirement that X should a normed space into a Banach space. This is the assumption in the source I have available. Furthermore, it seems fairly clear that completeness is needed. For example, let P be the normed space of all polynomials on [-1/2,1/2] with the uniform norm. The completion of P is the Banach space C[-1/2,1/2]. Let f : [0,1] → P be defined by f(0; x)=0, and

$f(t;x)={\frac {1-x^{\lfloor 1/t\rfloor }}{1-x}}.$

Then f is the uniform limit of step functions in P, but it fails to have a right limit at 0 (in P). Instead its right limit is in C[-1/2,1/2]:

$\lim _{t\to 0^{+}}f(t;x)={\frac {1}{1-x}}$

uniformly in x. silly rabbit (talk) 13:56, 10 May 2008 (UTC)