Writing a function inside the main method - Java

When Java 8 comes out, the Closure/Lambda functionality should make it so that you can define the max method in the main method. Until then, you'll only be able to define a method in the main method in special circumstances.

As it happens, your question does fall into the special circumstance. There is an interface (Comparable) which encapsulates the logic of comparing two things of the same type. As a result, the code could be rewritten as follows:

public class TestMax {
  public static void main(String[] args) {
    int i = 5;
    int j = 2;
    Comparator<Integer> compare = new Comparator<Integer>() {
        @Override
        public int compare(Integer o1, Integer o2) {
            // Because Integer already implements the method Comparable,
            // This could be changed to "return o1.compareTo(o2);"
            return o1 - o2;
        }
    };
    // Note that this will autobox your ints to Integers.
    int k = compare.compare(i, j) > 0 ? i : j;
    System.out.println("The maximum between is " + k);
  }
}

This only works because the comparator interface already exists in the standard Java distribution. The code could be made better through the use of libraries. If I were writing this code, I would add Google Guava to my classpath. Then I could write the following:

public class TestMax {
  public static void main(String[] args) {
    int i = 5;
    int j = 2;
    // The 'natural' ordering means use the compareTo method that is defined on Integer.
    int k = Ordering.<Integer>natural().max(i, j);
    System.out.println("The maximum between is " + k);
  }
}

I suspect that your question was more about the abilities of the Java language rather than standard practices having to do with ordering numbers (and other things). So this might not be useful, but I thought I'd share just in case.

No, you can't declare a method inside another method.

If you look closely at the code you provided it is just a case of bad formatting, the main method ends before the max method is declared.