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") …

Oct 15, 2016 · A continuous function is a function where the limit exists everywhere, and the function at those points is defined to be the same as the limit. I was looking at the image of a …

May 10, 2019 · Of course, the CDF of the always-zero random variable $0$ is the right-continuous unit step function, which differs from the above function only at the point of discontinuity at $x=0$.

Jan 27, 2014 · To understand the difference between continuity and uniform continuity, it is useful to think of a particular example of a function that's continuous on $\mathbb R$ but not …

Jan 13, 2012 · the difference is in definitions, so you may want to find an example what the function is continuous in each argument but not jointly

Nov 14, 2012 · Closure of continuous image of closure Ask Question Asked 13 years ago Modified 13 years ago

Proving the inverse of a continuous function is also continuous Ask Question Asked 12 years, 1 month ago Modified 8 years ago

Then the theorem that says that any continuous function on a compact set is uniformly continuous can be applied. The arguments above are a workaround this.

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?

Oct 29, 2017 · I believe it follows from the fact that we showed $f+g$ is continuous whenever $f$ and $g$ are continuous. Indeed, if $g$ is continuous, then $-g$ is clearly continuous.