Assignment:
QA). Let M be the set of functions defined on [0,1] that have a continuous derivative there ( one-sided derivatives at the endpoints).
Let p(x,y) = max_[0,1]|x'(t) - y'(t)|.
1).Show that ( M,p) fails to be a metric space.
2). Let p(x,y) = |x(0) - y(0)| + max_[0,1]|x'(t) - y'(t)|. Is (M,p) now a metric space?
QB). Let M be the set of continuous functions on [0,1] and define p(x,y) = integral from 0 to 1 of |x(t) - y(t)|dt. Does this define a metric space?
Provide complete and step by step solution for the question and show calculations and use formulas.