In Betrand Russell's 1908 essay "Mathematical Logic as Based on the Theory of Types" (page 227) he states a definition of continuity different from the canonical one:
"We call $f(x)$ continuous for x=a if, for every positive number $\sigma$ different from 0, there exists a positive number $\epsilon$, different from 0, such that for all values of $\delta$ which are numerically less than $\epsilon$, the difference $f(a+\delta)-f(a)$ is numerically less than $\sigma$."
This definition adds in an extra variable, $\sigma$ which does not appear in the usual definition of continuity. Why does he add in this variable? Was it the common definition at the time of publication (109 years ago), or was he using something that was uncommon at the time? I have tried looking to see if Russell had a particular definition of continuity which was different that others, but couldn't find anything.
Russell's essay was not primarily about this definition (it is philosophical), so not much is said about it in the essay, but its difference from the normal continuity was intriguing.
I think this is just the standard modern definition. What Russell calls $\sigma$ is $\epsilon$ today, his $\epsilon$ is our $\delta$ and his $\delta$ is what is sometimes $h$.
"Numerically less than" is probably "smaller in absolute value".
There may be a typo in your last line, which should be $$ f(a+\delta) =- f(a). $$