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Claptrap actually can...
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...climb stairs. I let my game sit at the main menu screen for a few minutes and it showed Claptrap in the background moving toward the bridge that spans the waterfall. There is a small flight of stairs at the beginning of the bridge and Claptrap actually climbs them to cross the bridge.
I like what I do, I do what I like I can quit and get it back like I'm ridin' a bike