if there is a small hole near the base of the knob that's what holds it in place. It fits over an pin on the part in the door, its spring loaded so press it in with a flat screwdriver and slide the knob on until it snaps into the hole on the knob.

I appreciate the reply, but there are no holes or anything that looks like a snap clasp.

What series lockset?

The handle looks like this

Here is a link that might be useful: Byron

http://consumer.schlage.com/Project%20Documents/P515-302-SH2-c.pdf

Here are install instructions for schlage locks note how you remove the knobs to switch them.

I have installed Schlage locks in my home and they work pretty much as others here have described. Never saw a pressed-on knob.
It probably took quite a yank to break and something is now bent or distorted badly. It might be best to just buy a new lockset. They are not super-expensive.
Be thankful your son was not seriously hurt
Best of luck.
Bernie

stash - thanks for the link, I found that online too. This is an interior set with no locks so the set up is different.

bondrey102: I've done door sets before too and I hadn't seen a pressed-on knob before either. I agree that the best fix is to just pick up a new knob. I don't know exactly how he managed it. He's 20 yo and luckily did not fall down the stairs.

FWIW I took out the two screws and the halves split in the middle with a knob and cover plate going to each side. For the short term, I banged the loose knob back on with a hammer. It's not straight and has some wiggle but at least it operates and stays on for now. I'll pick up a new knob set one of these days.

Thanks to all who posted for the input.

The document toward the end has instructons for interior door latch sets.

http://consumer.schlage.com/Project%20Documents/T00-0000R14-F.pdf

Dexter is made by Schlage and the knob sets are fixed and are put together with 2 screws.