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] &rarr; 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)