Q:

Does this affect the fact that $f$ is defined with several variables

Let us consider the function $f:ℝ^{2m+3q-1}→ℝ$ with $2m+3q-1$ variables. We know that the behavior of functions with several variables is different from those of one variable. My question is: Is it possible to restrict the analysis to one variable to see some proprties of the function on one axe and consider the other variables constants. Does this affect the fact that $f$ is defined with several variables.

A:

Yes, this is in fact what one tries to do. Functions of one variable have many peculiar properties which simplify their management (most of them depending on the fact that $\mathbb R$ is an ordered set). All these properties can be used in function of several variables once they are restricted to a single variable.
For example the function $f(x,t)=tx^4$ can be considered as a function of $t$ with $x$ fixed: $g(t) = t x^4$ or as a function of $x$ with $t$ fixed: $h(x)=tx^4$. The derivative $g'(t) = x^4$ is different from the derivative $h'(x) = 4tx^3$ and one needs some new notation to express this (namely partial derivatives):
$$
\frac{\partial f}{\partial t}(x,t) = g'(t) = x^4\\
\frac{\partial f}{\partial x}(x,y) = h'(x) = 4tx^3.
$$
In many situations, however, one uses multiple variables to express a multidimensional quantity. For example the position of a point in the plane can be expressed by two coordinates $(x,y)$ and a function which has a position as its input can be expressed as a function of two variables: $f(x,y)$. In this case, however, the two variables are to be considered as two "projections" of a single variable, the position. What one finds is that in such cases properties like continuity or differentiability cannot be expressed by simply requiring the corresponding properties on the two variables taken separately, instead must be expressed in terms of the pair of variables. For example the function
$$
f(x,y) = \begin{cases}
\frac{xy}{x^2+y^2} & \text{when $(x,y)\neq(0,0)$}\\
0 & \text{when $(x,y)=(0,0)$}
\end{cases}
$$
is continuous in both variables $x$ and $y$, but it is not considered a continuous function because there are points arbitrarily close to $(0,0)$ where the function is not close to the value $f(0,0)=0$ (namely: $f(t,t) = 1/2$). This is what makes the study of functions of several variables not trivial at all!
So the point is: functions of one variable are very important and you use them all the time when you want to study functions of several variables. However functions of several variables are somewhat more complicated than that and require the understanding of very interesting new topics.

