V - DIFFERENTIABLE FUNCTIONS

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This chapter discusses the differentiable functions. From elementary calculus, one knows that the sum, difference, and product of differentiable functions are differentiable; and with a little care to avoid zero in the denominator and even roots of negative numbers, the quotient, powers, and roots of differentiable functions are differentiable. In the case of transcendental functions, however, the situation is somewhat different. The definitions of trigonometric, logarithmic, exponential, hyperbolic, and inverse trigonometric functions all involve, in some way, the idea of a limit of algebraic functions; and, in fact, this is an essential feature of transcendental functions. The chapter discusses that C(1) [a, b] is complete under the norm of that space. Thus, to conclude that the limit of a sequence of continuously differentiable functions has a continuous derivative, it must be determined not only that the sequence of functions converges uniformly, but also that the sequence of derivatives converges uniformly.