n. the standard notation used in the definitions of limits, continuity, and related concepts; the principal notion is that a function tends to a limit at a given point if its value lies within some small ɛ (epsilon) of the limit whenever the independent variable is within a small δ (delta) of the given argument. Formally, a real function f(x) has a limit L at a point p, at which it is defined, if it satisfies the condition:
for every ɛ > 0 there exists a δ > 0 such that
| f(x) - L | < ɛ for all x such that | x - p | < δ.
More generally, a function has a limit at a point p in a metric space at which it is defined if it satisfies the same condition where the absolute differences | x - p | and | f(x) - L | are replaced by the corresponding metric distances, or in neighborhood notation, if
for every ɛ > 0 there exists a δ > 0 such that
f(x) ∈ N(ɛ, L) for all x such that x ∈ N(δ, p).
In each case, the function is then said to be continuous at p if its limit at p exists and equals f(p).