#Haskell, 52 bytes
f x=and$g(/=)$g(>)x
where g h y=zipWith h(tail y)y
I suspect I could get this a chunk smaller if I managed to get rid of the "where" construct, but I'm probably stuck with zipWith.
This works by making a list of the rises (True) and falls (False), then making a list of if the ajacent entries in this list are different
This is my first attempt at one of these, so I'll go through my thought process in case I've gone horribly wrong somewhere.
###Ungolfed Version (168 bytes)###
isBumpy :: [Char] -> Bool
isBumpy input = and $ areBumps $ riseFall input
where
riseFall ax@(x:xs) = zipWith (>) xs ax
areBumps ax@(x:xs) = zipWith (/=) xs ax
###Shorten names, remove type information (100 bytes)###
f x = and $ g $ h x
where
h ax@(x:xs) = zipWith (>) xs ax
g ax@(x:xs) = zipWith (/=) xs ax
###Move h into the main function soas it isn't defined in two placesis only used once (86 bytes)###
f ax@(x:xs) = and $ g $ zipWith (>) xs ax
where
g ax@(x:xs) = zipWith (/=) xs ax
###Realise that areBumps and riseFall are similar enough to abstract (73 bytes)###
f x = and $ g (/=) $ g (>) x
where
g h ya@(y:ys) = zipWith h ys ya
###Note that (tail y) is shorter than ya@(y:ys) (70 bytes)###
f x = and $ g (/=) $ g (>) x
where
g h y = zipWith h (tail y) y
###Tidy up; remove unneeded spaces (52 bytes)###
f x=and$g(/=)$g(>)x
where g h y=zipWith h(tail y)y
