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- https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/14%3A_Differentiation_of_Functions_of_Several_Variables/14.02%3A_Limits_and_ContinuityWe have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a func...We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.
- https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_v2_(Reed)/14%3A_Differentiation_of_Functions_of_Several_Variables/14.02%3A_Limits_and_ContinuityWe have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a func...We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.
- https://math.libretexts.org/Bookshelves/Analysis/A_Primer_of_Real_Analysis_(Sloughter)/04%3A_Topology_of_the_Real_Line/4.02%3A_Open_SetsLet \(x \in \bigcap_{i=1}^{n} U_{i} .\) Then \(x \in U_{i}\) for every \(i=1,2, \ldots, n .\) For each \(i\), choose \(\epsilon_{i}>0\) such that \(\left(x-\epsilon_{i}, x+\epsilon_{i}\right) \subset ...Let \(x \in \bigcap_{i=1}^{n} U_{i} .\) Then \(x \in U_{i}\) for every \(i=1,2, \ldots, n .\) For each \(i\), choose \(\epsilon_{i}>0\) such that \(\left(x-\epsilon_{i}, x+\epsilon_{i}\right) \subset U_{i} .\) Let \(\epsilon\) be the smallest of \(\epsilon_{1}, \epsilon_{2}, \ldots, \epsilon_{n} .\) Then \(\epsilon>0\) and \[(x-\epsilon, x+\epsilon) \subset\left(x-\epsilon_{i}, x+\epsilon_{i}\right) \subset U_{i}\] for every \(i=1,2, \ldots, n .\) Thus \[(x-\epsilon, x+\epsilon) \subset \bigcap…
- https://math.libretexts.org/Courses/Mission_College/Math_4A%3A_Multivariable_Calculus_(Kravets)/03%3A_Functions_of_Several_Variables/3.03%3A_Limits_and_ContinuityWe have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a func...We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.
- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/14%3A_Differentiation_of_Functions_of_Several_Variables/14.03%3A_Limits_and_ContinuityWe have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a func...We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.
- https://math.libretexts.org/Under_Construction/Purgatory/MAT-004A_-_Multivariable_Calculus_(Reed)/03%3A_Functions_of_Several_Variables/3.03%3A_Limits_and_ContinuityWe have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a func...We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.
- https://math.libretexts.org/Courses/University_of_Maryland/MATH_241/03%3A_Differentiation_of_Functions_of_Several_Variables/3.03%3A_Limits_and_ContinuityWe have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a func...We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/Interactive_Calculus_Q3/05%3A_Differentiation_of_Functions_of_Several_Variables/5.03%3A_Limits_and_ContinuityWe have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a func...We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/03%3A_Differentiation_of_Functions_of_Several_Variables/3.02%3A_Limits_and_ContinuityWe have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a func...We have now examined functions of more than one variable and seen how to graph them. In this section, we see how to take the limit of a function of more than one variable, and what it means for a function of more than one variable to be continuous at a point in its domain. It turns out these concepts have aspects that just don’t occur with functions of one variable.