The definition of continuity is:
$f$ is continuous at $a$ if: Given any $\epsilon>0 $,
$\exists \delta > 0$ st. $|x-a|<\delta \implies |f(x)-f(a)|< \epsilon$
$\delta$ obviously depends on the given $\epsilon $ ($\delta=\delta(\epsilon))$ and the range of values $x $ can take depends on $\delta$ but,
am I right in thinking that $\delta$ does not depend on $x$?