UNIFORM CONTINUITY AND DIFFERENTIABILITY

PRESENTED BY

PROF. BHUPINDER KAUR

ASSOCIATE PROFESSOR

GCG-11, CHANDIGARH

DEFINITION OF UNIFORM CONTINUITY

A function f is said to be uniformly continuous in an interval [a,b], if

given:

Є > 0, З δ > 0 depending on Є only, such that

|f(x1) – f (x2) < Є

Whenever x1, x2 Є [a,b] and |x1- x2|< δ

THEOREM

If f is uniformly continuous on an interval I, then it is continuous on I.

NOTE:

I is any interval, open or closed or semi open.

Converse of this Theorem need not be true.

Uniform continuity => continuity.

THEOREM

If f is continuous in closed interval I = |a,b| then f uniformly continuous in [a,b].

EXAMPLE

Show that function f(x) = 1/x is not uniformly continuous in (0,1].

SOLUTION

If possible suppose f is uniformly continuous in (0,1].

Given Є = ¼ > 0, З δ > 0 depending on Є only, s.t.

|f(x1) – f (x2) < Є = ¼ for x1, x2 in (0,1] and |x1 – x2| < δ

Let x1 = δ/2 and x2 = δ

Therefore |f(x1) –f (x2)| = |1/x1 – 1/x2|=

|2/ δ - 1/ δ| = 1/ δ

1/ δ < ¼ => δ > 4 => x2 > 4, not possible because then x2 (0,1]

Therefore our supposition is wrong => f is not uniformly continuous in (0,1].

NOTE: Continuity on closed interval => uniform continuity.

DEFINITION OF DERIVATIVE

Let f : [a,b] R be a function and C Є (a,b), then f’ is said to derivable or

differentiable at c, if

The limit in case it exists is called the derivative of f at c and is denoted by f’ (c)

NOTE:

f is derivable in open interval (a,b) is derivable at every point c of (a,b).

If f is derivable at c then f is continuous at c.

Geometrically f’ (c) represents the slope of the tangent to the curve y = f (x) at the point (c,f(c).

cx

cfxf

cx

)()(lim

THE CHIAN RULE (DERIVATIVE OF COMPOSITE OF TWO FUNCTIONS)

THEOREM

Statement: If f and g be no functions such that

Range of f is subset of Domain of g

f is derivable at c

g is derivable at f (c)

Then composite function g of is derivable at c and (gof)’(c)=g’(f(c)). f’(c).

DARBOUX’S THEOREM FOR DERIVATIVE

If f is a function defined and derivable on a closed interval [a,b] and f’ (a) and f’(b) and opposite sign i.e. f’ (a) f’(b) < 0, then there exist some point c Є(a,b) such that f’(c) = f’(c) = 0

Darboux intermediate value theorem for derivatives

Statement: If f is deriable in [a, b] and f’ (a) = f’ (b) and k is a number lying between f’(a) and f’(b) then there exist some c in (a,b) such that f’ (c) = k.

Example of Chain Rule

Let y = tan-1 (cosh x); put u = cosh x so that

y = tan-1u

y = tan-1 => dy/du = 1/1+u2 = 1/1+cosh2x

u = cosh x => du/dx = sinh x

By Chain Rule, dy/dx = dy/du, du/dx =

(1/1+cosh2x) (sinh x)

Problems based on Darboux Theorem

1. If f’(x) 0 for all x in (a,b) for all x in (a,b), then f’ (x) retains the same sign positive or negative in (a,b).

2. If f is defined and derivable on an interval I, then the range of f’ is either an interval or a singleton.

3. If f is defined and derivable on [a,b] such that f(a) = 0 = f (b) and f’ (a); f’(b) are of same sign, then show that f must vanish at least once in (a,b).

4. If f is derivable on [a,b] such that f(a) = 0 = f (b) and f (x) 0 for any x in (a,b) then prove that f’ (a) and f’ (b) must be of opposite signs.

5. If f is derivable at a point c then | f | is also derivable at c, provided f (c) 0 but converse may or may not be true.