Suppose I have the following function:
g = function(x) x+h
Now, if I have in my environment an object named h, I would not have any problem:
h = 4
g(2)
## should be 6
Now, I have another function:
f = function() {
    h = 3
    g(2)
}
I would expect:
rm(h)
f()
## should be 5, isn't it?
Instead, I get an error
## Error in g(2) : object 'h' not found
I would expect g to be evaluated within the environment of f, so that the h in f will be bound to the h in g, as it was when I executed g within the .GlobalEnv. This does not happen (obviously). any explanation why? how to overcome this so that the function within the function(e.g. g) will be evaluated using the enclosing environment?
There's a difference between the enclosing environment of a function, and its (parent) evaluation frame.
The enclosing environment is set when the function is defined. If you define your function g at the R prompt:
g = function(x) x+h
then the enclosing environment of g will be the global environment. Now if you call g from another function:
f = function() {
    h = 3
    g(2)
}
the parent evaluation frame is f's environment. But this doesn't change g's enclosing environment, which is a fixed attribute that doesn't depend on where it's evaluated. This is why it won't pick up the value of h that's defined within f.
If you want g to use the value of h defined within f, then you should also define g within f:
f = function() {
    h = 3
    g = function(x) x+h
    g(2)
}
Now g's enclosing environment will be f's environment (but be aware, this g is not the same as the g you created earlier at the R prompt).
Alternatively, you can modify the enclosing environment of g as follows:
f = function() {
    h = 3
    environment(g) <- environment()
    g(2)
}
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