qwuICp-05

# qwuICp-05 - 12 SPRING 2012 5 Partial Derivatives continued...

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12 SPRING 2012 5. Partial Derivatives, continued The extra condition on f that provides a converse for Proposition 4.5 is the continuity of its partial derivatives, in the sense that x °→ D j f i ( x )shou ldbecon t inuou sf rom U R n into R for each i, j . (See Proposition 5.3.) Defnition 5.1. A function f : U R n R m is continuously di f erentiable if f is di f er- entiable at each x U ,andth emap x °→ f ° ( x )f rom U into L ( R n , R m )i scon t inuou s . The set of continuously di f erentiable functions from U into R m is denoted C 1 ( U, R m ), and the expression “ f is C 1 ”isanabbrev iat ionforthestatemen t f C 1 ( U, R m ). C 1 ( U, R )is abbreviated to C 1 ( U ). Continuous di f erentiability is a stronger condition than di f erentiability, even for real- valued functions of a single real variable. Example 5.2. DeFne f : R R by f ( x )= ° x 2 sin 1 x if x ± =0 0i f x =0 . Then f ° ( x )ex istsforeach x , but the function x °→ f ° ( x ) is not continuous. Proposition 5.3. A function f : U R n

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qwuICp-05 - 12 SPRING 2012 5 Partial Derivatives continued...

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