If your friend goes through a difficult time, do you feel bad for him, or do you feel badly?
Correct answer: You feel bad for him.
This is one of those cases where the incorrect grammar is so widely used that the correct construction sounds wrong. Or maybe people feel like the longer word is... fancier, kind of like how one might say utilise instead of use.
Quick primer: Bad is an adjective; badly is an adverb. Adjectives modify nouns, and adverbs modify verbs, so you maybe you had a bad cup of coffee this morning and then behaved badly when someone cut you off in traffic.
So since feel is a verb, shouldn't it be badly? Nope, and here's why: Feel is a linking verb. Let's go the dictionary.
Linking verbs are not like regular action verbs. They function only to connect the subject of a sentence or clause with words that describe or identify that subject. And those words are either adjectives (or adjective phrases) or nouns (or noun phrases).
There are a number of linking verbs in English, among them be, become, seem, and all of the sensory verbs: smell, look, taste, sound, and feel.
You might say, for example, these cookies taste delicious! You wouldn't say these cookies taste deliciously. That would imply the cookies are doing the tasting, which, if you're tripping, ok, but it doesn't really work in day-to-day life.
As Grammar Girl points out, linking verbs "describe emotions or states of being," and require more sensitive handling than your average action verb. "To be" is the queen of all linking verbs, and you don't say, "I am badly" after you eat the cookies, you say "I am bad."
If you were to feel badly, you would be doing the act of feeling -- literally touching/groping someone or something -- badly. I'm having trouble thinking of a good example sentence, but here's my best shot: In the pitch dark, I felt my way badly around the room. But even that I probably wouldn't write, because it sounds awkward.
Here's where it gets tricky: Many linking verbs are also action verbs. So after a bout with the flu, you might wake up one morning and say "I feel strong today!" But you might feel strongly about whether to brush first or floss first.
"Feel badly," Merriam-Webster points out, is an example of grammatical hypercorrection -- people so want to get it right that they yank the grammatical wheel too hard in the other direction. (A rival dictionary reminds us that "between you and I" is another example of hypercorrection.)
Is that sort of clear? I would feel bad if it weren't.