continuous,
adj. 1a. (of a function) informally, having a value that changes gradually as the independent variable or variables change, so that at every value, a, of the independent variable, the difference between f(x) and f(a) approaches zero as x approaches a. More formally, a real function y = f(x) is continuous at a point a if and only if it is defined at x = a and
,
that is, precisely, if
for every ɛ > 0 there exists a δ > 0 such that
| f(x) - f(a) | < ɛ for all x such that | x - a | < δ.
A function is continuous on the left at the point if the above condition holds only for values of x less than a, and continuous on the right if it holds for values greater than a; it is continuous at a point if and only if it is continuous both on the left and on the right at that point. The function itself is said to be continuous if it is continuous at all points. The function is said to be uniformly continuous on a set if the value of δ depends only on ɛ and not on the point a in the set.
b. (of a curve) representing a continuous function. 2. (of a function defined between metric spaces) having the analogous property that y = f(x) is continuous at a point p if and only if it is defined at x = p and
for every ɛ > 0 there exists a δ > 0 such that
d(f(x), f(p)) < ɛ for all x such that d(x, p) < δ,
or in neighborhood notation, if
for every ɛ > 0 there exists a δ > 0 such that
f(x) ∈ N(ɛ, f(p)) for all x such that x ∈ N(δ, p).
If for all p in some set, the value of δ depends only on ɛ and not on the specific point p, the function is said to be uniformly continuous on the set. Every continuous function defined on a compact set is uniformly continuous thereon. 3. (of a function f between topological spaces, at a point p) more generally, having the property that, given any neighborhood V of f(p), there exists a neighborhood U of p with f(U) inside V. A function is then continuous at every point exactly if the inverse image of any open set is open (and of every closed set is closed). This reduces to the previous definition when the topological spaces are metric. See also limit, differentiable. 4. (Statistics) (of a random variable or random vector) not discrete; having a continuum of possible values so that its distribution requires integration rather than summation to determine its cumulative probability. 5. (of a measure or measure ring) another word for non-atomic.
,for every ɛ > 0 there exists a δ > 0 such that
| f(x) - f(a) | < ɛ for all x such that | x - a | < δ.
A function is continuous on the left at the point if the above condition holds only for values of x less than a, and continuous on the right if it holds for values greater than a; it is continuous at a point if and only if it is continuous both on the left and on the right at that point. The function itself is said to be continuous if it is continuous at all points. The function is said to be uniformly continuous on a set if the value of δ depends only on ɛ and not on the point a in the set.
b. (of a curve) representing a continuous function. 2. (of a function defined between metric spaces) having the analogous property that y = f(x) is continuous at a point p if and only if it is defined at x = p and
for every ɛ > 0 there exists a δ > 0 such that
d(f(x), f(p)) < ɛ for all x such that d(x, p) < δ,
or in neighborhood notation, if
for every ɛ > 0 there exists a δ > 0 such that
f(x) ∈ N(ɛ, f(p)) for all x such that x ∈ N(δ, p).
If for all p in some set, the value of δ depends only on ɛ and not on the specific point p, the function is said to be uniformly continuous on the set. Every continuous function defined on a compact set is uniformly continuous thereon. 3. (of a function f between topological spaces, at a point p) more generally, having the property that, given any neighborhood V of f(p), there exists a neighborhood U of p with f(U) inside V. A function is then continuous at every point exactly if the inverse image of any open set is open (and of every closed set is closed). This reduces to the previous definition when the topological spaces are metric. See also limit, differentiable. 4. (Statistics) (of a random variable or random vector) not discrete; having a continuum of possible values so that its distribution requires integration rather than summation to determine its cumulative probability. 5. (of a measure or measure ring) another word for non-atomic.