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I was in Wal-Mart the other day, and I decided I wanted to get something St. Patrick's Day-esque. They had this Hallmark pins which were pretty cute. On the back of the little piece of card the button is attached to is that warning:

CAUTION: CONTAINS FUNCTIONAL SHARP POINT

I just thought it was hilarious. I wonder if Hallmark sells anything with a dysfunctional sharp point.

And here I was clicking on this node thinking that I was going to find a warning not to attempt to differentiate a function which is continuous but is not smooth in its first derivative, eg: f(x) = abs(x*ln(x2)) in a neighborhood around x=0 looks like:



                         .      |     .
                                |
                                |    
                            .   |   . 
                                |    
                                |   
                              . | .     
                                |     
                                |
                               .|.
                                |
                                .
                 --------------------------------

Which many calculus textbooks call a function with a "sharp point". The limit of f'(x) from the left is minus infinity, and from the right is plus infinity, so from an analysis standpoint this is the sharpest point you can have. Be careful!

I leave it to Hallmark as an exercise to come up with other math warnings: CAUTION: NON-REMOVABLE DISCONTINUITY, etc.

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