Post on 22-Aug-2020
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.