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1. ### Continuity Problem

Well, I understand how the property f(x+y)=f(x)+f(y) is used to demonstrate f(0)=0 and obviously it is employed to expand f(a+h)=f(a)+f(h). I understand that this step is supposed to demonstrate the continuity of all a, but then why is h->0, and how exactly is continuity of a demonstrated if a...
2. ### Continuity Problem

Homework Statement Suppose that f satisfies f(x+y) = f(x) + f(y), and that f is continuous at 0. Prove that f is continuous at a for all a. Homework Equations f(x+y) = f(x) + f(y) Limit Definition Continuity: f is continuous at a if the limit as x approaches a is the value of the...
3. ### Basis for Polynomial Subspace

Ohh. I think I follow; by plugging in for a, the elements we obtained form a basis? That is, in each case of {(x2-x), (x3-x), (x4-x)}, 0 and 1 are roots: e.g., x2-x = x(x-1).
4. ### Basis for Polynomial Subspace

Thanks for the quick reply, btw. I think I catch your meaning. That means the coefficients must sum to 0, for every element in the subspace(?). Given your helpful reply, I don't feel inclined anymore to fix x(x-1) as an element. That is, the fact that 0 is a root follows from there being...
5. ### Basis for Polynomial Subspace

Homework Statement Let P_4(\mathbb{R}) be the vector space of real polynomials of degree less than or equal to 4. Show that {{f \in P_4(\mathbb{R}):f(0)=f(1)=0}} defines a subspace of V, and find a basis for this subspace. The Attempt at a Solution Since P_4(\mathbb{R}) is...