@Studentmath We have a theorem that gives us three different conditions that are all equivalent. But the general definition is as follows:
@Studentmath If $(V, ||.||_V), (W, ||.||_W)$ are normed vector spaces, then a map $f:V \rightarrow W$ is said to be continuous at $x \in V$ if $\forall \epsilon > 0, \exists \delta > 0 : \forall y : ||x-y||_V < \delta \implies ||f(x)-f(y)||_W < \epsilon$.
Suppose that $$M=\begin{pmatrix} 1 & 0 & 0 & . & . & 0 \\ 0 & 2 & 0 & 0 & . & . \\ 0 & 0 & . & 0 & . & . \\ . & 0 & . & . & . & . \\ . & . & . & . & . & 0 \\ 0 & . & . & . & 0 & n \end{pmatrix}$$
We note that $C(M)$ the subspace formed by the matrices which commute with M.
I proved that $C(M)$ ...
I am trying to understand the proof of the Borsuk-Ulam theorem for $S^2$ given in Hatcher's "Algebraic Topology" (Th. 1.10), as another person does here, but we are stuck at different places, so I hope this question is not considered a duplicate.
Borsuk-Ulam: If $f:S^2\rightarrow\mathbb R^2$ ...
Note the simple fact the integrand is positive and
$$\int_{0}^{1}\dfrac{1}{x\ln{x}}\cdot \underbrace{\frac{x}{\log(1-x)}}_{\large\text{near 0 it behaves like $-1$}}dx\longrightarrow \infty$$
Q.E.D.
Absolute values in calc 1 kill me, I don't even know how I'd draw this:
Find the volume of the solids you get by rotating the following graph around the x-axis
$$f(x)=(1+x^2)^{-1/2}, |x| \le 1$$
