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 with the phrase "ap calculus". The reason for using "ap calculus" instead of just "calculus" is to ensure that advanced stuff is filtered out.
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 piecewise continuous
12 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 decimal) t = number of years A = amount after time t The above is specific to continuous compounding.
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 $x=0$.
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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 $\mathbb R$.
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...
Usually when saying this, textbooks assume the so called infinity type of discontinuity, which apply precisely to points where a function is not defined and tends to infinity. I do understand 1/x is continuous on (0,infty) if you mean that, but I wouldn’t say it is false to say that as a function on R it has an infinity type discontinuity at ...
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 ...
9 Continuous Functions are not Always Differentiable. But can we safely say that if a function f (x) is differentiable within range $ (a,b)$ then it is continuous in the interval $ [a,b]$ . If so , what is the logic behind it ?
