incorrigibly

/ɪnˈkɔrɪdʒəbli/
adverb
  1. In a way that is impossible to correct, improve, or change; hopelessly or persistently.
    • He is incorrigibly optimistic, always seeing the bright side even in disaster.
    • She remains incorrigibly stubborn, refusing to admit she made a mistake.
    • The puppy was incorrigibly mischievous, stealing socks no matter how many times we scolded him.