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Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that for all $x_0\in\mathbb{R}$ we have $\lim\limits_{x\to x_0}f(x)=g(x_0)\in \mathbb{R}$.

Is $g$ a continuous function?

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    Have you researched on the web for Continous functions. Here is a good link: http://en.wikipedia.org/wiki/Continuous_function2012-03-16
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    By considering the constant sequence you directly have $f=g$ and then you have the definition of continuous2012-03-16
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    @Listing: Constant sequences (and, more generally, sequences $(x_n')$ such that $x_n' = x_0$ for some $n$) are likely to be excluded. There was some discussion about this definition some time ago, although I don't remember in which topic. Hence, the arguments one has to use are more sophisticated, and $f$ may no be continuous.2012-03-16
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    See ALso: http://math.stackexchange.com/questions/3777/is-there-a-function-with-a-removable-discontinuity-at-every-point2012-03-16
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    That is in fact the very definition of continuous!2016-06-16

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