Display back button on action bar

well this is simple one to show back button

actionBar.setDisplayHomeAsUpEnabled(true);

and then you can custom the back event at onOptionsItemSelected

case android.R.id.home:
this.finish();
return true;

I think onSupportNavigateUp() is the best and Easiest way to do so, check the below steps. Step 1 is necessary, step two have alternative.

Step 1 showing back button: Add this line in onCreate() method to show back button.

assert getSupportActionBar() != null;   //null check
getSupportActionBar().setDisplayHomeAsUpEnabled(true);   //show back button

Step 2 implementation of back click: Override this method

@Override
public boolean onSupportNavigateUp() {  
    finish();  
    return true;  
}

thats it you are done
OR Step 2 Alternative: You can add meta to the activity in manifest file as

<meta-data
        android:name="android.support.PARENT_ACTIVITY"
        android:value="MainActivity" />

Edit: If you are not using AppCompat Activity then do not use support word, you can use

getActionBar().setDisplayHomeAsUpEnabled(true); // In `OnCreate();`

// And override this method
@Override 
public boolean onNavigateUp() { 
     finish(); 
     return true; 
}

Thanks to @atariguy for comment.