The purpose of this paper is to examine which classes **F** of
functions from **R**^{n} into **R**^{m}
can be topologized in a sense that
there exist topologies \tau_{1} and \tau_{2} on
**R**^{n} and
**R**^{m}, respectively,
such that
**F** is equal to the class C(\tau_{1},\tau_{2})
of all continuous functions
f:(**R**^{n},\tau_{1})-->(**R**^{m},\tau_{2}).
We will show that the Generalized Continuum Hypothesis GCH
implies the positive answer for this question for a large
number of classes of functions **F** for which the sets
{x: f(x)=g(x)} are small in some sense for all different
f and g in **F**. The topologies will be Hausdorff and connected.
It will be also shown that in some model of set theory ZFC with GCH
these topologies could be completely regular and Baire.
One of the corollaries of this theorem is that
GCH implies the existence of a connected Hausdorff
topology **T** on **R**
such that the class **Lin** of all linear functions g(x)=ax+b
coincides with C(**T**,**T**).
This gives an affirmative answer to a question of Sam Nadler.
The above corollary remains true for
the class **P** of all polynomials,
the class **A** of all analytic functions
and the class of all harmonic functions.

We will also prove that several other classes of real functions
cannot be topologized. This includes
the classes of C^{infinity} functions, differentiable functions,
Darboux functions, and derivatives.

**Last modified April 24, 1999.**