The first
"graph" is a function f
having a positive derivative at 0, but not increasing in any
neighbourhood of 0.

The second
"graph" shows that there is no knight's tour on a
4-by-4 chessboard.
(In case you were wondering, a formula
for f is: f(0)=0,
f(x)=x^{2}sin
(1/x) +x/2 if x=0.)