Continuity

In mathematics, the notion of continuity of a function relates to the idea that the "value" of the function should not jump abruptly for any vanishingly "small" variation to its argument. Another way to think about a continuity of a function is that any "small" change in the argument of the function can only effect a correspondingly "small" change in the value of the function.

Formal definitions of continuity
We can develop the definition of continuity from the $$\delta-\epsilon$$ formalism which is usually taught in first year calculus courses to general topological spaces.

Function of a real variable
The $$\delta-\epsilon$$ formalism defines limits and continuity for functions which map the set of real numbers to itself. To compare, we recall that at this level a function is said to be continuous at $$x_0\in\mathbb{R}$$ if (it is defined in a neighborhood of $$x_0$$ and) for any $$\varepsilon>0$$ there exist $$\delta>0$$ such that
 * $$ |x-x_0| < \delta \implies |f(x)-f(x_0)| < \varepsilon. \,$$

Simply stated, the limit
 * $$\lim_{x\to x_0} f(x) = f(x_0).$$

This definition of continuity extends directly to functions of a complex variable.

Function on a metric space
A function f from a metric space $$(X,d)$$ to another metric space $$(Y,e)$$ is continuous at a point $$x_0 \in X$$ if for all $$\varepsilon > 0$$ there exists $$\delta > 0$$ such that


 * $$ d(x,x_0) < \delta \implies e(f(x),f(x_0)) < \varepsilon . \,$$

If we let $$B_d(x,r)$$ denote the open ball of radius r round x in X, and similarly $$B_e(y,r)$$ denote the open ball of radius r round y in Y, we can express this condition in terms of the pull-back $$f^{\dashv}$$


 * $$f^{\dashv}[B_e(f(x),\varepsilon)] \supseteq B_d(x,\delta) . \, $$

Function on a topological space
A function f from a topological space $$(X,O_X)$$ to another topological space $$(Y,O_Y)$$, usually written as $$f:(X,O_X) \rightarrow (Y,O_Y)$$, is said to be continuous at the point $$x \in X$$ if for every open set $$U_y \in O_Y$$ containing the point y=f(x), there exists an open set $$U_x \in O_X$$ containing x such that $$f(U_x) \subset U_y$$. Here $$f(U_x)=\{f(x') \mid x' \in U_x\}$$. In a variation of this definition, instead of being open sets, $$U_x$$ and $$U_y$$ can be taken to be, respectively, a neighbourhood of x and a neighbourhood of $$y=f(x)$$.

Continuous function
If the function f is continuous at every point $$x \in X$$ then it is said to be a continuous function. There is another important equivalent definition that does not deal with individual points but uses a 'global' approach. It may be convenient for topological considerations, but perhaps less so in classical analysis. A function $$f:(X,O_X) \rightarrow (Y,O_Y)$$ is said to be continuous if for any open set $$U \in O_Y$$ (respectively, closed subset of Y ) the set $$f^{-1}(U)=\{ x \in X \mid f(x) \in U\}$$ is an open set in $$O_x$$ (respectively, a closed subset of X).