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I am Looking for a multivariable function with an assymptote and a gradient bounded away from zero.

For example: let $f:\mathbb{R}^k\rightarrow\mathbb{R}$ be a function such that

  1. $lim f(x_1,\cdots x_k)\rightarrow M$ where $M$ is a finite constant
  2. $\forall x\in \mathbb{R}^k$, $\nabla f>\delta(x)$ where $\delta(x)>0$.
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  • $\begingroup$ What do you mean by an asymptote in a multivariable function? Anyway, if you want $\lim_{|\mathbf{x}| \to \infty} f( \mathbf{x} ) = M < \infty$, then necessarily the gradient must approach to $0$, so that such a function cannot exist. $\endgroup$ – Crostul Oct 30 '16 at 12:53
  • $\begingroup$ Is $f(\mathbf{x})=\sum_{i=1}^kx_i$ not such a function? It has a constant gradient, not equal to $0$, and is its own asymptote. $\endgroup$ – ahorn Oct 30 '16 at 13:01
  • $\begingroup$ I am looking for a function that is never " too close" to the assymptote, while by defn. this distance (i.e., the gradient) will approach zero, I want it to "reach" 0 only in the limit... $\endgroup$ – MoRkO Nov 1 '16 at 9:33

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