# How do we know if a function is infinitely continuous?

How do we know if a function is infinitely continuous? That mean it is continuous at every point?

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We prove it. What's the function? – Qiaochu Yuan May 10 '12 at 19:40
A function $f:A\to B$ is said to be continuous on $A$ if for every $\varepsilon>0$, there is a $\delta>0$ such that $|x-a|<\delta$ implies $|f(x)-f(a)|<\varepsilon$ for all $x,a\in A$. – EsX_Raptor May 10 '12 at 19:42
If by "infinitely continuous" you are refering to the symbol $\mathcal{C}^{\infty}$, this means that at each point, the function has derivatives of all orders; in particular, it is continuous and differentiable everywhere, the derivative is continuous and differentiable at every point , the second derivative is continuous and differentiable at every point, etc. – Arturo Magidin May 10 '12 at 19:42
@QiaochuYuan - Is there any process or algorithm that is for every function on the proof? Or just we have to arrange the usage of theorem or order of theorem for different theorem? – Victor May 10 '12 at 19:44
@Victor: no. As with many problems, it's possible to encode undecidable problems (e.g. the Halting problem) into the problem of whether some arbitrary function is continuous. In practice these artificial examples don't come up and generally speaking you use simple established facts about continuous functions. – Qiaochu Yuan May 10 '12 at 19:49

Going with the definition of continuity of a function $f : D \to \mathbb R$ at a point $x_0$ is a starting point : $$\forall \varepsilon > 0, \exists \delta > 0 \quad s.t. \forall x \in D, \quad |x-x_0| < \delta \quad \Rightarrow \quad |f(x)-f(x_0)| < \varepsilon$$ but in general one wants to use the fact that most elementary functions are continuous (polynomials, rational polynomials where the denominators don't vanish, fractional exponents, sines, cosines, exponentials, log, and sums/products/compositions of those) because working with the definition all the time will make you crazy. But it's good to be able to work with it though.