# Is the exponential function continuous in R^n? How can one show this? Let's say for a function like f:R^2->R,f(x,y)=x^y.

Is the exponential function continuous in ${\mathbb{R}}^{n}$? How can one show this? Let's say for a function like $f:{\mathbb{R}}^{2}⟶\mathbb{R},f\left(x,y\right)={x}^{y}$
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Johnny Parrish
The domain is not the entire ${\mathbb{R}}^{2}$, as we have trouble with negative values of x. If x>0, we have that ${x}^{y}={e}^{y\mathrm{log}x}$, which is continuous because the one-variable exponential and $\left(x,y\right)↦y\mathrm{log}x$ are continuous.