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February 14th, 2016, 06:29 AM   #1
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for all x close to c

What is the precise mathematical meaning of the phrase "for all $\displaystyle x$ close to $\displaystyle c$"? Does it mean that for all $\displaystyle x$ that satisfy the inequality $\displaystyle |x-c|<\epsilon$, where $\displaystyle \epsilon$ is a small positive number? This phrase is not only used in the informal definition of limit but also used in the definition of decreasing and increasing functions.
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February 14th, 2016, 03:44 PM   #2
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In general, yes. However to be precise, the context must be given.
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February 14th, 2016, 05:27 PM   #3
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Generally, I think it tends to mean that $x$ is sufficiently close to $c$ that there is nothing exciting happening between the two, such as a discontinuity. Sometimes, your more formal definition is required which suggests that $|x-c|$ is genuinely small.
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