trying to recall the $\epsilon, \delta$ definition of continuity, I came up with the following:
A function is continuous at $x$ if $\forall \epsilon > 0 \; \exists \; \delta > 0: |f(x-\delta) - f(x+\delta)| < \epsilon$.
This is very likely not equivalent to the Weierstrass' definition of continuity at $c$:
$\forall \epsilon > 0 \; \exists \; \delta > 0: |x-c| < \delta \Rightarrow |f(x) - f(c)| < \epsilon $.
Could you please point out where the first statement fails to be equivalent to Weierstrass'?