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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 Any help would be welcome. 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 have probably misinterpreted something I was looking at the image of a piecewise continuous
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 $\mathbb r$.
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 Closure of continuous image of closure ask question asked 12 years, 11 months ago modified 12 years, 11 months ago A map is continuous if and only if for every set, the image of closure is contained in the closure of image So continuously differentiable means differentiable in a continuous way.
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. Is the derivative of a differentiable function always continuous My intuition goes like this
If we imagine derivative as function which describes slopes of (special) tangent lines to points on a.
This would mean that the derivative of a function is always continuous on the domain of the function, but i have encountered counterexamples
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