Continuous function

A function is continuous at a point when its limit there equals its value.

Bounded continuous functions on a topological space form a Banach space with the supremum norm. A uniform limit is continuous and bounded, proving completeness. This space differs from the bounded uniformly continuous functions: free-transport composition is jointly continuous in time and position, but need not be continuous in the global sup norm as time varies.

A real or extended-real function is upper semicontinuous when every strict sublevel set is open. Equivalently its negative is lower semicontinuous. An arbitrary infimum of continuous functions is upper semicontinuous, even when the infimum is over an uncountable family; its strict sublevel sets are unions of the corresponding open sublevel sets.

If $|h(x)|\le C$ near $c$ and $a(x)\to 0$ as $x\to c$, then $a(x)h(x)\to 0$ by the squeeze theorem, even if $h$ has no limit. Defining the product to be zero at $c$ gives continuity there. For example, $\sqrt{|x|}\sin (1/\sin x)$ extends continuously at zero; the same oscillation near a nonzero multiple of $\pi$ has nonvanishing amplitude and does not have a continuous zero extension.

A function is lower semicontinuous when its strict superlevel sets are open, equivalently when its sublevel sets are closed. In Euclidean spaces this is the sequential lower semicontinuity condition $f(x)\le {\mathrm{liminf}}_{k}f(x_k)$ whenever $x_k\to x$. This closedness property is essential to existence of proximal minimizers.

An extended-real functional $f$ is sequentially lower semicontinuous when $x_n\to x$ implies $f(x)\le {\mathrm{liminf}}_{n}f(x_n)$.

A sublevel set consists of points whose objective value is at most a chosen threshold. Coercivity makes finite sublevel sets bounded, while sequential lower semicontinuity makes them sequentially closed sets in the selected topology.

On a metric space, a functional is sequentially lower semicontinuous exactly when every sublevel set $\{x:f(x)\le \eta \}$ is closed.

A function is locally constant when every point has a neighbourhood on which the function is constant.

If $f:[a,b]\to \mathbb{R}$ is continuous and $y$ lies between $f(a)$ and $f(b)$, then some $c\in [a,b]$ satisfies $f(c)=y$.

A continuous injective real-valued function on a real interval is strictly monotone. A value outside the interval between two endpoint values would, by the intermediate value theorem, create a repeated value. Applying this observation to subintervals gives consistent strict ordering. If the left endpoint value is smaller than the right endpoint value, the function is strictly increasing; otherwise it is strictly decreasing.

If $f:[0,1]\to \mathbb{R}$ is continuous and $f(0)=f(1)$, then for every positive integer $n$ there is $\alpha \in [0,1-1/n]$ with $f(\alpha +1/n)=f(\alpha )$. For $n>1$, the continuous difference $h_n(x)=f(x+1/n)-f(x)$ satisfies ${\sum }_{j=0}^{n-1}{h}_n(j/n)=0$. Either one sampled difference is zero or two have opposite signs; the intermediate value theorem then supplies a zero between them. For $n=1$, the endpoint chord suffices.

If a continuous real-valued function on a real or complex vector space has opposite signs at two points, restricting it to the line segment between them and applying the intermediate value theorem gives a zero.

If a continuous $g:[0,na]\to \mathbb{R}$ satisfies $g(na)-g(0)=na$, then the incrementssatisfy ${\sum }_{k=0}^{n-1}h(ka)=0$. Continuity forces $h$ to vanish somewhere on $[0,(n-1)a]$.

Every continuous injective function from a real interval to the real line is strictly monotone. A continuous bijection $\mathbb{R}\to \mathbb{R}$ therefore has a continuous inverse.

If $f$ is continuous at $x$ and $g$ is continuous at $f(x)$, then $g\circ f$ is continuous at $x$.

The continuity set of a function is the set of points where it is continuous. For a real function $f$, define its oscillation at $x$ byThe sets $\{x:{\omega }_f(x)<\epsilon \}$ are open, and $f$ is continuous at $x$ exactly when ${\omega }_f(x)=0$. Hence its continuity set is the $G_{\delta }$ Borel set

A function $f$ on the real line is right-continuous at $t$ when $f(s)\to f(t)$ as $s\downarrow t$.

A càdlàg function is right-continuous and has a finite left limit at every positive time. The name abbreviates the French phrase /continue à droite, limites à gauche/.

A function is uniformly continuous when one input tolerance works at every point of its domain.

A nonnegative uniformly continuous function on $[0,\infty )$ with finite integral tends to zero. Otherwise uniform continuity gives disjoint intervals of a fixed width on which the function exceeds a fixed positive value, contradicting integrability. This supplies a useful dissipation argument for Lyapunov functions with merely continuous vector fields.