Hetware wrote: > On 10/13/2013 1:18 AM, Arturo Magidin wrote:
>> >> In particular, functions are not "declared" to be continuous, they >> are not "assumed" to be continuous, and they are not "asserted" to be >> continuous. > > That's nonsense. In variational calculus we assert the existence of an > infinite number of continuous, twice differentiable functions which > coincide at the boundaries of the parameter domain, but are otherwise > arbitrary. > > A statement such as "let M be a smooth manifold", asserts the existence > of a continuous function.
That is quite different. You have a function f defined by a formula, and more than once you have asserted that it is continuous. If f's continuity matters to you, you _prove_ its continuity. f is indeed continuous at all points in its domain, but it isn't continuous at 3; it's not even defined there.
-- The world will little note, nor long remember what we say here Lincoln at Gettysburg