text: Next: Properties Up: Functions, continuity and differentiability Previous: Functions, continuity and differentiability
  
  ## Definitions
  
  The study of differentiable functions is the study of functions that mimic the behaviour of polynomials approximately''. To begin with we must formally define the notion of approximation.
  
  Exercise 34   For any real number 0 < x < 1 show that xn is a decreasing sequence with limit 0.
  
  In particular, we see that a polynomial that vanishes to order (n + 1) at 0 satisfies the following condition on functions of one variable.
  
  Definition 1   A function g(x) of one variable is said to be in o(xn) if for any > 0 there is a > 0 so that
  
  | g(x)| < | xn| for all x so that | x| < .
  
  An alternative notion is
  
  Definition 2   A function g(x) one variable is said to be in O(xn) is there is a > 0 and a constant C so that
  
  | g(x)| < C| xn| for all x so that | x| <
  
  Clearly, any polynomial that vanishes to order n is O(xn). Further, it is clear that an function g(x) that is O(xn) is o(xn - 1) and any function that is o(xn) is O(xn).
  
  We can extend these notions to many variables as well. A function g(x1,..., xn) of n variables is said to be in o(xn) (respectively O(xn)) if for all lines (x1,..., xn) = (xc1,..., xcn) through the origin the restricted function f (x) = g(xc1,..., xcn) is in o(xn) (respectively O(xn)). We can further extend this to define o((x - b)n) and O((x - b)n) where b = (b1,..., bn) is some point, as a way of approximating functions near this point.
  
  We say that g and f agree upto o((x - b)n) (or f approximates g upto o((x - b)n)) if f - g is in o((x - b)n). Note in particular, that f and g take the same value at b.
  
  A function is differentiable n times at the point c if it is approximated upto o((x - b)n) by a polynomial (of degree n). Clearly, a polynomial of any degree is differentiable by the results of the previous section. In the one variable case we write this as follows
  
  f (x) = a0 + a1x + ... + anxn + o((x - c)n)
  
  Exercise 35   Show that for any two functions f and g in o(xn) and a function h which is differentiable n times at the origin, the function h . f + g is in o(xn).
  
  Exercise 36   Show that the numbers ak are uniquely determined by the function f.
  
  Now the number a1 depends on f and the point c. Now suppose that f is differentiable (1 times) at all points c so that it can be written as above near every point c. Then we can define the derived function f' by letting f'(c) = a1 for each point c; the function f' is also called the derivative of f. Now it clear that if f is the function given by a polynomial P then f' is dP/dx. Thus we also use the notation df /dx for f'. We have the derivation property as well.
  
  Exercise 37   If f, g and h are differentiable then so is hf + g and
  
  (hf + g) = f + h +
  
  However, unlike the condition of vanishing to order n at c, the condition o((x - c)n) is not very well behaved.
  
  Exercise 38   Show that f (x) = x2sin(1/x) is o(x) but the derivative of f' is not o(x0).
  
  A function f (x) is called continuous at a point c if f (x) - f (c) is o(x - c) (i. e. it is differentiable 0 times!). It is called continuous it it has this property at all points. Thus we would like to study functions f which are differentiable and in addition the derivative f' is continuous. Such functions are provided by the fundamental theorem of calculus.
  
  Next: Properties Up: Functions, continuity and differentiability Previous: Functions, continuity and differentiability
  Kapil H. Paranjape 2001-01-20
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