A function is "differentiable" if it has a derivative. A function is "continuous" if it has no sudden jumps in it. Until today, I thought these were merely two equivalent definitions of the same c...

6 days ago · This is well defined and continuous since $\gamma$ is simple and $\theta (a)=\theta (b)$. So $\log f (z)$ is a continuous choice of the logarithm on $\gamma ( [a,b])$.

Does a continuous mapping have to map the boundary to the boundary Ask Question Asked 7 years, 10 months ago Modified 3 years, 3 months ago

In general, is a bounded linear operator necessarily continuous (I guess the answer is no, but what would be a counter example?) Are things in Banach spaces always continuous?

To find examples and explanations on the internet at the elementary calculus level, try googling the phrase "continuous extension" (or variations of it, such as "extension by continuity") simultaneously …

Prove the absolute value function of a continuous function is continuous [duplicate] Ask Question Asked 12 years, 1 month ago Modified 11 years, 4 months ago

Clearly if $f$ is continuous then its kernel is closed set. for the converse, assume that $f\neq0$ and that $f^{-1}(\{0\})$ is a closed set. Pick some $e$ in $X$ with $f(e)=1$.

In basic calculus an analysis we end up writing the words "continuous" and "differentiable" nearly as often as we use the term "function", yet, while there are plenty of convenient (and even fairly precise) …

You'll find topological properties with indication of whether they are preserved by (various kinds of) continuous maps or not (such as open maps, closed maps, quotient maps, perfect maps, etc.). For …

Then, is this measure uniquely determined? I know if I tell you how to integrate all measurable functions, then this measure is of course uniquely determined. Because integrate characteristic functions will …