Which is name of this theorem? real valued $f(x)$ continuous and compact-supported $\Rightarrow f(x)$ bounded: $|f(x)| < \infty$??
It is always true? (Non linear case?): I want to know if it apply to continuous but nowhere-differentiable functions as the Wiener Processes: which is continuous, but given that it has non-zero probability of jump to infinity, and its derivative is irrestricted (actually is undefined since they have unbounded variation and finite quadratic variation), in principle it is possible to have an infinite rate of change, but in every finite extension sections of the Wiener Process it never happens...
Is the theorem of the main question the reason that keeps these continuous nowhere-differentiable functions bounded?
If $f(x) \in \mathbb{C}$, The theorem still holds for complex functions or not?
I have seen it referred to as the Boundedness Theorem, but I suspect it doesn't have a truly universal name. It is an immediate consequence of the stronger (and more well-known) Extreme Value Theorem: continuous functions on compact intervals achieve both a minimum and a maximum.
This is an application of another fundamental fact of topology, without a name as far as I know: the continuous image of a compact set is compact. Of course, Heine-Borel Theorem and/or Bolzano-Weierstrass Theorem have a part to play here, but this fundamental fact of topology means that we can consider continuous functions on compact subsets of $\Bbb{R}$ or $\Bbb{C}$, and they will map to compact subsets of $\Bbb{R}$ or $\Bbb{C}$. That is, the images will be both bounded and closed, achieving a maximum and minimum if they lie in $\Bbb{R}$, and a maximum/minimum modulus if they lie in $\Bbb{C}$.
This applies to any continuous functions, regardless of continuity properties. It definitely does apply to continuous, nowhere-differentiable functions. I don't know if I'd call it "the reason" that continuous, nowhere-differentiable are bounded, but it does prove that they will always have this property.