## Lectures on the Theory of Functions of Real Variables, Volume 1 |

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algebraic function called cells consider containing points continuous function converges uniformly corollary corresponding cubical division curve defined denote difference quotient differential coefficient discrete aggregate Dist domain of definition equation example exists f(xx f(xy finite number finite or infinite fix the ideas frontier points gives graph Hence hypothesis improper integrals indeterminate forms infinite discontinuity infinity of points integrable in 21 integrand interval 21 inverse function Let f(x limited and integrable limiting point minimum monotone obviously one-valued function oscillation partial derivatives point aggregate points of 21 points of infinite positive integer positive number prove rational function rational numbers respect Rolle's theorem satisfied segments sequence simply regular singular integrals relative singular point small at pleasure suppose tangent theorem totally differentiable uniformly convergent uniformly evanescent upper content vanish variables

### Popular passages

Page 48 - ... the product of two positive numbers is positive, the product of a positive and a negative number is negative and the product of two negative numbers is positive again.

Page 46 - This proves the first part of the theorem. The second part is proved similarly.

Page 395 - We might find и by integrating first with respect to x and then with respect to у ; this process would be indicated Ъy the equation (œ,y) dydx.

Page 208 - A contained in the switching surface is a sliding mode domain if for each e > 0 there exists a 8 > 0 such that...

Page iii - Some of the chapters deserve especial mention on account of their clearness and finish. For example, it would be hard to find more attractive accounts of the theory of implicit functions, indeterminate forms, and the definition and properties of a definite integral than are given in Chapters IX, X...

Page 189 - Chapter 2 that the limit of the sum of two functions is the sum of the limits, and the limit of the product is the product of the limits...

Page 19 - Lectures on the Theory of Functions of Real Variables" by Professor Pierpont of *Yale University we find the following statement : "Division by zero is excluded in modern mathematics. The admission of division by zero by the older mathematicians, Euler for example, has caused untold confusion. We shall see it is entirely superfluous.

Page 354 - A have content, it is necessary and sufficient that for each positive number e there...

Page 247 - Ь, f(x) must first increase, and then decrease ; or first decrease, and then increase ; hence f'(x) must change from + to — , or from — to +, and therefore, if continuous, pass through 0.99.

Page 118 - A symbol which takes on more than one value, in general an infinity of values, is a...