Recent content by corey115

You have been more than helpful! I feel like I have a much greater understanding of this proof now!

Since that is just a constant (scalar in our case), it would equal c times whatever the limit of just f(x) was.

For the 2nd item you listed, would the additive inverse of f be -f and the additive identity of f be the 0 function that is discontinuous where ever f is discontinuous? As for the 3rd item, I'm not entirely sure how I would go about showing that. It makes sense intuitively as to why it would...

Closure under scalar multiplication. Commutative/Associative for addition/multiplication. Additive identity/inverse. Multiplicative identity. Distributive property. Associative property for scalar multiplication.

Would that not be just the sum of the two limits of f and g at x, or if one or both of those two functions is discontinuous at x, then I would just use the whatever sided-limit that I need?

It will also be continuous because the sum of two continuous functions will also be continuous. But do I need extra arguments besides that because V is the set of piecewise continuous functions?

Homework Statement A function f:[a,b] \rightarrow ℝ is called piecewise continuous if there exists a finite number of points a = x0 < x1 < x2 < ... < xk-1 < xk = b such that (a) f is continuous on (xi-1, xi) for i = 0, 1, 2, ..., k (b) the one sided limits exist as finite numbers Let V be the...

Homework Statement Let F be an infinite field (that is, a field with an infinite number of elements) and let V be a nontrivial vector space over F. Prove that V contains infinitely many vectors. Homework Equations The axioms for fields and vector spaces. The Attempt at a Solution...