A continuous function is a function whose output remains arbitrarily close to its value at a point whenever the input is sufficiently close to that point. Continuity is defined relative to the function’s domain and the structures used to describe closeness. For real-valued functions, it is expressed through limits or an epsilon–delta condition; in more general spaces, through open sets. It is a central concept in mathematical analysis and topology. (jirka.org)
Continuity at a point
Let (D) be a subset of the real numbers, and let (f:D\to\mathbb R). The function is continuous at (a\in D) if, for every (\varepsilon>0), there exists a (\delta>0) such that [ x\in D,\quad |x-a|<\delta \quad\Longrightarrow\quad |f(x)-f(a)|<\varepsilon. ] Here (\varepsilon) specifies an allowed output deviation, while (\delta) specifies a sufficiently small input deviation. The choice of (\delta) may depend on both (\varepsilon) and (a). A function is continuous on its domain if it is continuous at every point of that domain. (jirka.org)
At an accumulation point (a) of (D), this is equivalent to the limit condition [ \lim_{\substack{x\to a\x\in D}}f(x)=f(a). ] For an interior point of an interval, both one-sided limits must equal the function value. At an included endpoint, only approaches within the domain matter. Every function is continuous at an isolated point of its domain, because a sufficiently small neighborhood contains no other domain points. (jirka.org)
Continuity also has a sequential characterization: (f) is continuous at (a) exactly when every sequence (x_n\in D) converging to (a) satisfies (f(x_n)\to f(a)). This equivalence holds in metric spaces, but preservation of convergent sequences alone does not characterize continuity in all topological spaces. (math.ucla.edu)
Examples and discontinuities
Every polynomial is continuous on (\mathbb R). Rational functions are continuous wherever their denominators are nonzero. Exponential, logarithmic, and trigonometric functions are continuous on their respective domains. Thus (x\mapsto1/x) is continuous on (\mathbb R\setminus{0}); strictly speaking, continuity at zero is not a property of this function because zero is outside its domain. (openstax.org)
Failures of continuity at domain points take several forms:
- A removable discontinuity occurs when a finite limit exists but differs from the assigned value. For example, setting (f(x)=x) for (x\ne0) and (f(0)=1) creates a removable discontinuity at zero.
- A jump discontinuity occurs when finite left and right limits exist but differ, as for the function equal to zero for (x<0) and one for (x\ge0).
- An infinite discontinuity involves unbounded behavior near the point.
- An oscillatory discontinuity can occur without unboundedness: assigning any value at zero to (\sin(1/x)), initially defined for (x\ne0), cannot make it continuous there. (openstax.org)
The familiar description “a graph that can be drawn without lifting a pencil” is therefore only an informal picture for certain interval-domain functions, not a general definition. (openstax.org)
Operations and principal theorems
Sums, differences, products, and scalar multiples of continuous real-valued functions remain continuous. A quotient remains continuous wherever its denominator is nonzero. Composition also preserves continuity, provided the component functions have compatible domains and codomains. (jirka.org)
The intermediate value theorem states that a continuous function on ([a,b]) assumes every value between (f(a)) and (f(b)). In particular, opposite signs at the endpoints guarantee at least one zero between them. The theorem establishes existence, not uniqueness. (jirilebl.github.io)
The extreme value theorem states that a continuous real-valued function on a nonempty compact domain attains both an absolute minimum and an absolute maximum. A closed, bounded real interval is such a domain. Without these hypotheses, attainment can fail: (f(x)=x) on ((0,1)) is bounded but reaches neither its infimum nor its supremum. (jirilebl.github.io)
In calculus, continuity connects differentiation and integration. Existence of a finite derivative implies continuity, but the converse fails: (|x|) is continuous yet not differentiable at zero. A continuous function on a closed, bounded interval has a Riemann integral. The fundamental theorem of calculus further gives [ \frac{d}{dx}\int_a^x f(t),dt=f(x) ] at interior points when (f) is continuous. (jirka.org)
Stronger forms and limits of functions
Uniform continuity requires one (\delta) for each (\varepsilon) that works simultaneously at every domain point. Ordinary continuity allows (\delta) to vary with the point. Every continuous function on a compact metric space is uniformly continuous. However, (f(x)=x^2) is continuous on (\mathbb R) without being uniformly continuous there. (jirka.org)
Lipschitz continuity imposes a quantitative bound, [ |f(x)-f(y)|\le L|x-y|, ] with a fixed finite constant (L\ge0). It implies uniform continuity, although the converse does not hold. For example, (\sqrt{x}) is uniformly continuous on ([0,1]) but not Lipschitz continuous on that interval. (jirka.org)
A uniform limit of continuous real-valued functions is continuous. Pointwise convergence alone is insufficient: the continuous functions (f_n(x)=x^n) on ([0,1]) converge to zero for (x<1) and to one at (x=1), producing a discontinuous limit. (jirka.org)
Metric and topological formulations
For a map between metric spaces, the epsilon–delta definition replaces absolute differences with the respective distance functions. This includes multivariable functions, where closeness concerns the entire input vector rather than separate coordinate approaches. (jirka.org)
For topological spaces (X) and (Y), a map (f:X\to Y) is continuous if the preimage (f^{-1}(U)) of every open set (U\subseteq Y) is open in (X). Continuity does not require images of open sets to be open. A bijective continuous map with a continuous inverse is a homeomorphism. Continuous maps preserve compactness: the image of a compact space is compact. (math.ucla.edu)