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Unformatted text preview: f : R ! R by: f ( x ) = ( e ± 1 =x 2 ; if x 6 = 0 ; if x = 0 It can be shown that f is di²erentiable in±nitely many times at x = 0. Determine the values of f (0) and f 00 (0). Problem 5: Let f;g be functions which are both di²erentiable at x = c . Moreover, assume that f ( c ) = g ( c ) = 0 and g ( c ) 6 = 0. Prove that: lim x ! c f ( x ) g ( x ) = f ( c ) g ( c ) (Note that this is not L’Hospital’s rule.) Problem 6: Let f be any function on the real line and suppose that: j f ( x ) ± f ( y ) j ² j x ± y j 2 for all x;y 2 R . Prove that f is a constant function....
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This note was uploaded on 03/18/2012 for the course MATH 67 taught by Professor Schilling during the Fall '07 term at UC Davis.
- Fall '07