21-356 Lecture 2

# On the other hand we have that f 0 y x2 0 if y 0

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Unformatted text preview: )￿ = and ￿ (xδ − 0)2 + (yδ − 0)2 = ￿ δ2 2 = δ √ 2 <δ ￿￿ ￿ ￿ ￿ δδ ￿ |f (xδ , yδ ) − f (0, 0)| = ￿f 2 , 2 − 0￿ = 1. This proves that f is not continuous in the origin. On the other hand, we have that f (0, y ) = x2 = 0 if y = 0, and f (0, y ) = 0 otherwise. So, f (0, y ) = 0 for every y ∈ R. Hence, the function y → f (0, y ) is continuous in y = 0. Analogously f (x, 0) = x2 , and, again, x ￿→ f (x, 0) is continuous in x = 0. Exercise 0.3. Let f : R2 → R be deﬁned as: ￿ xy if (x, y ) ￿= (0, 0) 2 2 f (x, y ) := x +y . 0 otherwise Is f continuous in the origin? Why? 3 Directional derivatives and Diﬀerentiability Let E ⊂ RN , let f : E → R and let x0 ∈ E . Given direction v ∈ RN , let L be the line through x0 in the direction v, that is, L := {x ∈ RN : x = x 0 + tv , t ∈ R } 1 and assume that x0 is an accumulation point of the set E ! L. The directional derivative of f at x0 in the direction v is deﬁned as !f f (x0 + tv) " f (x0 ) (x0 ) := lim , t!0 !v t provided the limit exists in R. In the special case in which v = ei , the directional !f derivative ! ei (x0 ), if it exists, is called the p...
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