Why is it impossible to build a compiler that can determine if a C++ function will change the value of a particular variable?

Question

I read this line in a book:

It is provably impossible to build a compiler that can actually determine whether or not a C++ function will change the val

Answer 1

Imagine such compiler exists. Let's also assume that for convenience it provides a library function that returns 1 if the passed function modifies a given variable and 0 when the function doesn't. Then what should this program print?

int variable = 0;

void f() {
    if (modifies_variable(f, variable)) {
        /* do nothing */
    } else {
        /* modify variable */
        variable = 1;
    }
}

int main(int argc, char **argv) {
    if (modifies_variable(f, variable)) {
        printf("Modifies variable\n");
    } else {
        printf("Does not modify variable\n");
    }

    return 0;
}