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how to prove it is a cusp!?

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/0:
SAY YOU HAVE THE GRAPHS

FOR

AND FOR

THE GRADIENT AT X = 0 IS THE SAME FOR BOTH GRAPHS, YET THAT POINT IS A CUSP................................

HOW DOES ONE GO ABOUT PROOFING THAT IT IS A CUSP???????????????????????????????????????????????????

Mao:
if

then f(x) a cusp at x=a

/0:
But the limits are the same aren't they?????????????????????????????????

Mao:
for a smooth join they are the same (hence differentiable)

but for a sharp turn they are two different values approaching from left and right.

/0:
But for the equation I put up, the equations are differentiable at x = O, and they approach the same values from left and right. HOWEVER, there is a cusp at x = O , so how does that work??????

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