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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 $\displaystyle xc<\epsilon$, 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 $xc$ is genuinely small.


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