qwuICp-05

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online