Proving Subspace of Continuous Functions with Integral = 0

[theFuture]

So for my LA class I am to prove that all functions f such that they are contiunous over the [0,1] and their integral over the same integral = 0 is a subspace of the function space of continuous functions over [0,1]. So I think my proof is fine but I have one semi-technical question. Is it ok just to state:

"if f, g are continuous over [0,1] f+g must also be by a theorem of calculus and if f is continuous rf must also be by a theorem of calculus."

I'm leaning towards no but I'm not sure

 

[HallsofIvy]

Yes, both of those statements are true. If both f and g are continuous at x= a, then $lim_{x->a}f(x)= f(a)$ and $lim_{x->a}g(x)= g(a)$. It follows then that $lim_{x->a}{f(x)+ g(x)}= f(a)+g(a)$ so that f+ g is continuous wherever f and g are.
Similarly, $lim_{x->a}rf(x)= r lim_{x->a}f(x)= rf(a)$ so that rf is continuous wherever f is.

 

[mathwonk]

i think he was asking if it was ok to assume those things in a linear algebra class which assumes a calculus course and i say yes.

those things are not really relevant to the problem however, but only background needed to show the problem is well posed. the essential poiint is the linearity properties of the integral.