I am trying to learn weak derivatives. In that, we call $\mathbb{C}^{\infty}_{c}$ functions as test functions and we use these functions in weak derivatives. I want to understand why these are called test functions and why the functions with these properties are needed. I have some idea about these but couldn't understand them properly.
Also, I'll be happy if any one can suggest some good reference on this topic and Sobolev spaces.
Suppose you want to find the solution of a differential equation, $f'' = gf$ for example.
Take any solution $f$ of this equation, then if you take any function $\psi \in \rm C^{\infty}_c$ it is true to write $$f'' = gf \Longrightarrow \psi f'' = \psi gf \Longrightarrow \int \psi f'' = \int \psi g f \Longrightarrow \int \psi'' f = \int \psi g f$$
Conclusion : any solution of the differential equation $f$ satisfies $\int \psi'' f = \int \psi g f$ but it is possible that functions that are only continuous satisfies the same equation ! These solutions will be called weak solutions because they are solutions of a weaker problem.
Now, why have we chosen the functions $\psi$ to be in $\rm C^{\infty}_c$ ? It is because we transfered the derivation operation from $f$ to $\psi$ by integration by part ; this integration by part goes well only if you suppose that $\psi$ has compact support. I encourage you to do the details of the last implication and it will become clearer.
To understand why we are calling them test functions, we have to understand what distributions are and where they come from.
Usually, to evaluate a function, we compute its value at the point where we want to know it. But remember that there are spaces of functions (or equivalence classes of functions) such as $L^p$ spaces where the value on a point is not a good representation of the underlying function (it may even have no sense at all). Then, why not trying to evaluate the function as some sort of a weighted mean ? And an integral of $f$ times a function wisely chosen can be seen as a weighted mean.
We define :
$$T_f(\phi)=\int_{\mathbb{R}}f(x) \phi(x) dx.$$
In fact, test functions are just a new way to "know" a function. Because we have the choice of the space from which we take our tests functions, why not taking one with very good properties ? So we do the choice of $\mathbb{C}^{\infty}_{c}$. By doing so, we can proceed to integrate a large variety of functions and we also have linearity and continuity (compared to the topology of $\mathbb{C}^{\infty}_{c}$) of the mapping $T_f$ when it has a meaning. We can also define a lot of useful operations by reporting every problem one would meet with $f$ on the tests functions.
We just have contructed a continuous linear application from $\mathbb{C}^{\infty}_{c}$ to $\mathbb{R}$. Those kind of applications are known as linear forms. So we define that a distribution is a continuous linear form over $\mathbb{C}^{\infty}_{c}$ to $\mathbb{R}$.
This way, we can take $\phi$ being narrower around the point we are considering with constant aera to know the "value" of $f$ near it.
Note that this is just the tip of the icerberg, and that the OP may have understood why we are calling them this way by now. But I wanted to share this POV on this site.
Specifically why it is called test function. (Essentially not different from another answer.)
Strichartz liked to introduce the concept with physical intuition.
Suppose you want to measure the temperature at a point in the room. You hold out your thermometer, but the tip of it is actually a blob of glass with some liquid inside, so are you really measuring the temperature at any actual point? At best it is a weighted average in a neighborhood of the point.
