What's the difference between continuous and piecewise …
2016年10月15日 · A piecewise continuous function doesn't have to be continuous at finitely many points in a finite interval, so long as you can split the function into subintervals such that each interval is continuous. A nice piecewise continuous function is the floor function: The function itself is not continuous, but each little segment is in itself continuous.
Proving the inverse of a continuous function is also continuous
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calculus - Is there a shorthand or symbolic notation for ...
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) shorthands for representing the latter, I'm not aware of a …
Continuity and Joint Continuity - Mathematics Stack Exchange
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What is a continuous extension? - Mathematics Stack Exchange
There are other ways a function can be a continuous extension, but probably the most basic way (and likely about the only way you'll see in elementary calculus) is that you have a function that is not defined at some point (maybe more than one point), but the limit of the function exists at that point(s), so if you simply define (like how you ...
real analysis - prove the integral function is continuous
Hint: the projection of a continuous function onto any coordinate is continuous, and the integral of a continuous real function is continuous. In a general sense, we may define the space $\mathbb{R}^2$ to be the collection of ordered pairs from $\mathbb{R}$, equipped with the property that the projection of any continuous function onto either ...
Difference between continuity and uniform continuity
2014年1月27日 · I understand the geometric differences between continuity and uniform continuity, but I don't quite see how the differences between those two are apparent from their definitions. For example, my book
elementary set theory - Is there such thing as a continuous set ...
2019年9月20日 · I've been studying preference continuity in microeconomics, and it seems obvious to me that preferences must be defined in a continuous set in order to be continuous themselves. But I think this is often left to one side as preferences are commonly thought in $ \mathbb{R}_+^L $ , which is a continuous set (However I am more concerned on the ...
is bounded linear operator necessarily continuous?
Added @Dimitris's answer prompted me to mention, beyond the fact that the implication on normed spaces indeed is an equivalence, that it's the converse which holds in the wider context of topological vector spaces, while the proposition mentioned here fails: there are bounded discontinuous linear operators, yet every continuous operator remains ...
real analysis - Proof that $f (x) = x^2$ is continuous ($\delta ...
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