For proving continuity of single variable functions of the form $\displaystyle f:\mathbb{R}\to\mathbb{R}$ there is the epsilon-delta method, but for multivariable functions of the form $\displaystyle f:\mathbb{R}^n\to\mathbb{R}$ where $\displaystyle n\ge 2$, we need to show that $\displaystyle \lim_{(x_1,...,x_n)\to (a,...,n)}f(x_1,x_2,...,x_n)=f(a,b,...,n)$ fromallpossible directions including curves in $\displaystyle \mathbb{R}^n$ that pass through (a,b,...,n) so how do we go about proving continuity?