"limit along a path" equivalent to usual definition of limit? At the institution where I teach, regrettably, we do not teach students the real definition of the limit in our calculus classes.  I am teaching a little complex analysis, and I would like to use the notion of $\lim_{z \to z_0} f(z)$, with at least a little rigor.  If I mention $\epsilon$ and $\delta$ my students will not understand it at all.  Is $\lim_{z \to z_0} f(z) =v$ equivalent to the assertion that $f(r(t)) \to v$ as $t:0\to 1$ along any path $r:[0,1) \to \mathbb{C}$ with $r(t) \to z_0$ as $t \to 1^-$?  My students might appreciate the idea of the pathwise definition.  It is clear to me that the standard, $\epsilon$-$\delta$ definition implies the pathwise criterion, but I embarassed to admit that I do not know if the converse is true.

Proof by contrapositive. Suppose that $\lim_{z\to z_0}f(z)=v$ is false. Then there is a sequence $z_n\to z_0$  such that $f(z_n)\not\to v$. Define a path $r:[0,1]\to\mathbb C$ as follows: $r(1)=z_0$,  $r(1-1/n)=z_{n}$ for each $n$, and $r$ is affine on each interval $[1-1/n,1-1/(n+1)]$. This is a continuous path ending at $z_0$, and  $f(r(t))\not\to v$ as $t\to 1-$.
