By Edwin Beckenbach

Collage of CaliforniaEngineering and actual sciences extension series.Includes bibliographies.Based at the Statewide lecture sequence on combinatorial arithmetic provided by means of the college of California, college Extension, Engineering and actual Sciences department, in 1962.

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Therefore, the 2 cosine takes all values between 1 and −1. This implies that exp(i · θ) is onto U. The goniometric addition theorem from proposition 210 in volume 1 is a consequence of the group homomorphism property of exp. For 0 ≤ θ < θ + η < π , it yields cos(θ + η) = cos(θ) cos(η) − sin(θ) sin(η) < cos(θ) cos(η) < cos(θ), 2 so the cosine function is strictly monotonously decreasing. So for every x ∈ 0, 1 , there is exactly one θ ∈ 0, π2 such that cos(θ) = x. By cos(θ)2 +sin(θ)2 = 1, the sine function is monotonously increasing from 0 to 1 as θ moves from 0 to π .

C 1 functions are also called continuously diﬀerentiable. Example 107 All polynomial functions as well as exp, sin, and cos are C ∞ functions. The function ⎧ ⎨ x2 f (x) = ⎩−x 2 if x ≥ 0, if x < 0 is continuous on R. Its derivative exists, is continuous, and is deﬁned by ⎧ ⎨ 2x if x ≥ 0, f (x) = ⎩−2x if x < 0. The second derivative f , however, does not exist at x = 0, where there is a jump from −2 to 2. Therefore f is in C 1 , but not in C 2 . Let us close with the very important mean value theorem.

Then there is δ > 0 such that 0 ≤ t < δ implies |f (ξ + t) − f (ξ) − t · f (ξ)| < t · f (ξ). But this means that f (ξ + t) > f (ξ), a contradiction. So taking x = ξ yields f (x) = 0. Fig. 4. Mean value theorem. 1 Partial Derivatives The existence of the derivative of a function is not evident in general, but there is an important additional information that gives necessary and suﬃcient conditions for the existence of derivatives. This information is provided by partial derivatives. The idea is completely natural: In order to understand the behavior of a function f on an open set U ⊂ Rn , one does not evaluate the function at every point of U but rather considers the restriction of f to special curves ci : Ui → U deﬁned on open sets Ui ⊂ R.