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I have problem understanding limit and contuinity of a multivariable function. Could someone give GEOMETRICAL interpretation of the meaning of limit and contuinity? What does it mean to say that a limit at a point exists in multivariable function/ or it does not exist?

Thank You

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What is your geometrical interpretation of a multivariable function? A function $\mathbb R^2 \to \mathbb R$ as a three-dimensional surface, perhaps? – Rahul Oct 23 '12 at 19:29
we can generalize for that case in this discussion. I mean a three dimensional surface would be sufficient. But if it is possible to give for n dimensional hyperspace, I would appreciate that too. I know algebraically, delta epsilon notation but would appreciate its geometric interpretation as well as what limit "geometrically" means. – 007resu Oct 23 '12 at 19:53
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A couple of examples might help ...

First take a function that maps from $\mathbb R$ to $\mathbb R^2$ or from $\mathbb R$ to $\mathbb R^3$. The function describes a parametric curve. If the function has a discontinuity it means (roughly) that there is an abrupt "jump" in the function value. In other words, there is a gap in the curve.

Next, take a function that maps from $\mathbb R^2$ to $\mathbb R^3$. The function describes a parametric surface. If the function has a discontinuity it means (roughly) that there is an "hole" in the surface.

As with familiar real-valued functions, a discontinuity doesn't necessarily imply a "jump"; it might correspond instead (I think) to a place where the function oscillates infinitely fast/often. But, for the purposes of intuition, I'd suggest you ignore those kinds of discontinuities, and focus on the "jump" ones.

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