Does c++11 lambda really support closure? There's a semantic conflict in function variable

I personally feel that C++11 lambda has some conflict with C/C++ function in that, a function local variable's life ends with function, but in FP, lambda is an object thus its variables has life cycle as long as the lambda.

I've got a small test

#include<stdio.h>
int main()
{
  auto f=[](int input){
    int local=3;
    return [=](int x){return input+local+x;};
  };
  auto f1=f(3);
  auto f2=f(4);

  printf("%d,%d\n",f1(2),f2(2));
  return 0;
}

g++ -std=c++11, it prints "8,9"

It's my expectation for FP, but for C language scope, its behavior should be "undefined", because both "input" and "local" dies after the "f" declaration.

So question:

For both input parameter and internal variable, does lambda object store them somewhere to make sure, they are still available after lambda definition? Or my test is undefined behavior?

Thanks.