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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 Since the sobolev space only cares about function up to a set of measure zero, we could ask questions about whether functions in the space are continuous, strongly differentiable, etc., but those questions are not invariant under modifications on a set of measure zero, so they can only be answered by seeing if there are sufficiently smooth. I was looking at the image of a piecewise continuous

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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 uniformly continuous on $\mathbb r$. Continuous from the left/right ask question asked 4 years, 5 months ago modified 4 years, 5 months ago This might probably be classed as a soft question

But i would be very interested to know the motivation behind the definition of an absolutely continuous function

To state a real valued function. Following is the formula to calculate continuous compounding a = p e^(rt) continuous compound interest formula where, p = principal amount (initial investment) r = annual interest rate (as a This is a general question A function is said to be continuous

Can it still have vertical asymptotes Looking at the definition of continuity, i would say no Proving the inverse of a continuous function is also continuous ask question asked 11 years, 11 months ago modified 7 years, 10 months ago 3 this property is unrelated to the completeness of the domain or range, but instead only to the linear nature of the operator

Yes, a linear operator (between normed spaces) is bounded if and only if it is continuous.

I know that the definition derives from calculus, but why do we define it like that?i mean what kind of property we want to preserve through continuous function?