Smallest number of contiguous monotonic subsequences
Brachylog, 12 bytes
~c:{&lt;=|&gt;=}al

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This returns false. for the empty list [].
Explanation
(?)~c                 Take a list of sublists which when concatenated result in the Input
     :{&lt;=|&gt;=}a        Each sublist must be either increasing or decreasing
              l(.)    Output is the length of that list

This will return the smallest one because ~c will generate choice points from the smallest number of sublists to the biggest.