How do we know if a function is infinitely continuous? That mean it is continuous at every point?
You shouldn't say "infinitely continuous" for this. A function is continuous if it is continuous at every point of its domain (that is the adopted definition in, say, real analysis).
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.
Hope that helps,
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