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 February 14th, 2016, 06:29 AM #1 Newbie   Joined: Jan 2014 Posts: 20 Thanks: 0 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.
 February 14th, 2016, 03:44 PM #2 Global Moderator   Joined: May 2007 Posts: 6,856 Thanks: 745 In general, yes. However to be precise, the context must be given.
 February 14th, 2016, 05:27 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,700 Thanks: 2682 Math Focus: Mainly analysis and algebra 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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