Why does initializing an extern variable inside a function give an error?

The reason defining an external variable inside a function does not make sense is the following:

When you declare a symbol extern, you are telling the compiler to link all such occurrences of this value into the same symbol. Any occurences of extern int i; in your program would link to the externally defined i. Look at this example:

#include <iostream>
using namespace std;

extern int i;
int i = 10;
void test()
{
    std::cout << "Hi" << i << std::endl;
}

int main()
{
    extern int i;
    i++;
    test();
}

This example should output hi11. HOwever, if we remove the extern inside main, it will output 10. This is because without extern, i is not linking to the global i, but creating it's own local copy of i.

The reason that defining an extern i inside a function does not make sense, is what if we allowed any function to "define" i. Which function runs first? When does it get defined?

Assume the following example to be valid, what would the output be???

#include <iostream>
using namespace std;

extern int i;
int i = 10;
void test()
{
    std::cout << "Hi" << i << std::endl;
}

void test2() {
    extern int i = 1000;
    std::cout<< "HI" << i << std::endl;
}

void test3() {
    extern int i;
    i = 1000;
    std::cout<< "HI" << i << std::endl;
}

int main()
{
    extern int i;
    i++;
    test();
    i = 0;
    test2();
}

Should the output of test2 be 0, or 1000? Also look at my test3, here we are concisely saying, link my i to the externally defined i, and assign it's value as 1000. This is very different from trying to "initialize" a value.

In short, extern variables really only make sense as globals, and should be defined in global scope. In your examples, the first version doesn't compile either for me. I find this interesting. It might be worth looking at the standards docs to see if this is defined concisely, or if your compiler might be handling this in a way designed to add additional protection...

By adding an initialiser to the declaration, it becomes a definition of the global variable. It's equivalent to the same definition without extern, which is what your book means when it says it "overrides the extern".

While global variables can be declared (using extern) inside a function, they cannot be defined there, only at namespace scope. That's why the second snippet is an error.

If you want to know why the designers of C (whence these rules came to C++) chose to allow declarations but not definitions here, then I'm afraid I don't know the language's history in enough detail to answer.