How to save edit form without closing it

You can do it by adding a button that will save the record without closing the edit form.

Below is examples for synchronous and asynchronous cases.

function on_edit_form_created(item) {
    var save_btn = item.add_edit_button('Save and continue');
    save_btn.click(function() {
        if (item.is_changing()) {
            item.post();
            try {
              item.apply();
            }
            catch (e) {
              item.alert_error(error);
            }
            item.edit();
        }
    });
}

function on_edit_form_created(item) {
    var save_btn = item.add_edit_button('Save and continue');
    save_btn.click(function() {
        if (item.is_changing()) {
            item.disable_edit_form();
            item.post();
            item.apply(function(error){
                if (error) {
                    item.alert_error(error);
                }
                item.edit();
                item.enable_edit_form();
            });
        }
    });
}