Is there melody in Jazz and Funk?

YES!  Perhaps someone had you listen to something really *out there* or obscure, like progressive jazz - but funk and and *most* categories of jazz are well aware of the melody - because after introducing it to you,they often improvise, so you can hear just how creative they can be with it.

In Jazz, yes, but catch it quick before it ends and the solos begin. In funk, yes, technically, but I can see how it would be hard to discern. In Herbie Hancock's Canteloupe Island, (funk jazz fusion?)  it's hard to tell if that is a melody, or just part of the groove. I think that would be in the eye of the listener to decide. I always ask myself that same question when I hear this. Another song is Moanin'. Is that a melody? A long groove intro? I think it was meant for us to decide. 

Absolutely.                                                         

Yes there is melody in Jazz. Perhaps the reason some people have a problem with it is because the main characteristic of Jazz is improvisation. The melody or a theme is introduced at the beginning and then the musicians take off.  Early Jazz was played in polyphonic style where each instrument plays one melody against another. Louis Armstrong was one of the first to play solo apart from the other musicians.

Plenty of contemporary jazz and fusion has no discernable melody, sure. I've listened to jazz for some decades and I still prefer to enjoy my music, not be challenged by it. Two words that mean pretty much the same in music but are used in different context - melody and theme. People tend to thing of melody as verse and often meant to be sung. Beethoven's 5th symphony has that "theme" - duh, duh, da DUH! Certainly recognizable as a lyrical thread. Much jazz is the same. Whether it is theme or melody, there is a recognizable thread. Structure. Not going to argue about pedantics.

There's a pro-melody trend in jazz these days that I find refreshing. A lot of musicians are digging into the Top 40 music of the classic rock days - taking the best things by such people as Sting, Stevie Wonder, and the Stones and recording jazz versions. Yes, this is *commercial* and that's not a dirty word. The fact that lots of people like something doesn't make it bad. 

Yes.