Lectures on the Theory of Functions of Real Variables, Volume 2

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Ginn, 1912 - Calculus - 645 pages
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Page 583 - Bis auf die neueste Zeit hat man allgemein angenommen, dass eine eindeutige und continuirliche Function einer reellen Veränderlichen auch stets eine erste Ableitung habe, deren Werth nur an einzelnen Stellen unbestimmt oder unendlich gross werden könne.
Page 583 - Werth nur an einzelnen Stellen unbestimmt oder unendlich gross werden könne. Selbst in den Schriften von Gauss, Cauchy, Dirichlet findet sich meines Wissens keine Äusserung, aus der unzweifelhaft hervorginge, dass diese Mathematiker, welche in ihrer Wissenschaft die strengste Kritik überall zu üben gewohnt waren, anderer Ansicht gewesen seien.
Page 454 - A contained in the switching surface is a sliding mode domain if for each e > 0 there exists a 8 > 0 such that...
Page 490 - Osc/= 0. xa 494. 1. jFor / to be supracontinuous at x = a, it is necessary and sufficient that for each e > 0, there exists a 8 > 0, such that Similarly the condition for infracontinuity is /(a) - e < f(x) , for any x in Fi(a).
Page 580 - Pierpont pointed out, the prototypical curve has the following properties: 1. It can be generated by the motion of a point. 2. It is continuous. 3. It has a tangent. 4. It has a length. 5. When closed, it forms the complete boundary of a region. 6. This region has an area. 7. A curve is not a surface. 8. It is formed by the intersection of two surfaces.
Page 582 - We not only say that the magnitude shall pass from one state to another by gradations imperceptible to our senses, but we also demand that between any two states another state exists and so without end.
Page 378 - L-integrable, it is necessary and sufficient that for each e > 0, there exists a separated division D of 31, for It is necessary.
Page 582 - METHOD OF LIMITS AS APPLIED TO TANGENCY. Euclid defines a tangent to a circle as a straight line which meets the circumference, but being produced, does not cut it : and from this definition he deduces the fundamental theorem that a tangent is perpendicular to the radius drawn to the point of contact.
Page 89 - P reduces to a0 if we set x = a, we see that every power series converges for at least one point. On the other hand, there are power series which converge at but one point, eg <z0 + i:Oa) + 2:<>-a)2 + 3!Oa)3+ ... (2 For if x=£ a, lim n!
Page 584 - It may be that we regard them as inextensible strings whose length is got by straightening them out. A less obvious way to measure their lengths would be to roll a straightedge over them and measure the distance on the edge between the initial and final points of contact.

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