Undying means something that never comes to an end — it will not die, no matter how much time passes. It goes before a noun: 'undying love', 'undying loyalty', 'undying gratitude', 'undying fame', 'my undying thanks'. What is worth noticing is which nouns it chooses. Almost always they are feelings and reputations: love, devotion, loyalty, gratitude, hatred, admiration, fame, glory. You do not say 'undying furniture' or 'undying rain'. The reason is in the picture: to say a feeling will never die is to treat it as something alive, something with a heart in it. That is a promise, not a measurement. So the word is emotional and a little grand. It belongs to poetry, songs, films, wedding speeches, and old novels — 'I swear my undying love' is the sort of line a character says on one knee. This has an important side effect for students: in everyday modern speech, using it seriously can sound over-the-top, and English speakers now often use it jokingly. 'You brought me coffee? You have my undying gratitude.' The joke works precisely because the word is so enormous next to a cup of coffee. Both uses are alive, and which one you are hearing depends entirely on tone and situation. Though actually, notice the shape of the word: it is built from 'dying', the -ing form, but 'undying' is not a verb and has no verb behind it. There is no 'to undie'. It is simply an adjective built out of that piece, meaning 'not dying, ever'.