calculus - Relation between differentiable,continuous and integrable ...
The containment "continuous"$\subset$"integrable" depends on the domain of integration: It is true if the domain is closed and bounded (a closed interval), false for open intervals, and for unbounded intervals.
Proof that the continuous image of a compact set is compact
I know that the image of a continuous function is bounded, but I'm having trouble when it comes to prove this for vectorial functions. If somebody could help me with a step-to-step proof, that would be great.
Why are norms continuous? - Mathematics Stack Exchange
Describe why norms are continuous function by mathematical symbols.
Showing that $\\ker T$ is closed if and only if $T$ is continuous.
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 $f (e)=1$.
What's the fastest way to tell if a function is uniformly continuous or ...
On the smaller closed interval the derivative is bounded; on the entire open interval the function does have vertical asymptotes and cannot be uniformly continuous. Re Dan Fisher's example of $\sqrt x …
Continuous versus differentiable - Mathematics Stack Exchange
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...
Continuous function proof by definition - Mathematics Stack Exchange
Continuous function proof by definition Ask Question Asked 12 years, 8 months ago Modified 6 years, 6 months ago
calculus - Are functions considered continuous at endpoints ...
In either case, a function is continuous on its domain if it is continuous at every point in the domain. Thus a function can be continuous on either $ [a,b]$ or $ (a,b)$.
Topological properties preserved by continuous maps
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 …
is bounded linear operator necessarily continuous?
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?