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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 x^{n} 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(
x^{n}) if for any
> 0 there is a
> 0 so that

g(
x) <

x^{n} for all
x so that 
x <
.
An alternative notion is
Definition 2
A function
g(
x) one variable is said to be in
O(
x^{n})
is there is a
> 0 and a constant
C so that

g(
x) <
C
x^{n} for all
x so that 
x <
Clearly, any polynomial that vanishes to order n is O(x^{n}). Further,
it is clear that an function g(x) that is O(x^{n}) is
o(x^{n  1})
and any function that is o(x^{n}) is O(x^{n}).
We can extend these notions to many variables as well. A function
g(x_{1},..., x_{n}) of n variables is said to be in
o(x^{n})
(respectively
O(x^{n})) if for all lines
(x_{1},..., x_{n}) = (xc_{1},..., xc_{n}) through the origin the restricted
function
f (x) = g(xc_{1},..., xc_{n}) is in o(x^{n}) (respectively
O(x^{n})). We can further extend this to define
o((x  b)^{n})
and
O((x  b)^{n}) where
b = (b_{1},..., b_{n}) 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) = a_{0} + a_{1}x + ^{ ... } + a_{n}x^{n} + o((x  c)^{n})
Exercise 35
Show that for any two functions f and g in
o(x^{n}) and a
function h which is differentiable n times at the origin, the
function
h^{ . }f + g is in
o(x^{n}).
Exercise 36
Show that the numbers a_{k} are uniquely determined by the function
f.
Now the number a_{1} 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) = a_{1} 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
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) = x^{2}sin(1/x) is o(x) but the derivative of f'
is not o(x^{0}).
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
20010120