Real Analysis Lecture ppt

7/23/2019 Real Analysis Lecture ppt

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Real Analysis

Lecture 6: Differentiation

Manasa Mandava

Indian School of Business, Hyderabad

Term 1, 2015


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Uniform continuity

Definition (Uniform continuity)

Let f :XY . The function f is uniformly continuous on X iffor every >0 there exists a >0 such that dY(f(p), f(q))<

for all p and q in X for which dX(p, q)< .

TheoremA function is continuous if it is uniformly continuous.


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Example : Uniformly continuous function

Example

Let f(x) =

x, x(0, ). Then for every >0 choose =2.Then for all x, yR such that|x y|< 2,

|f(x) f(y)|2 =|x y|2 |x y||x+ y|=|x y|< 2.

Thus, it follows from the above inequality that|f(x) f(y)|< for all x, yR such that|x y|<

2

. Since the choice =2

foreach does not depend on x, y, the function f(x) =

x is

uniformly continuous.


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xamp e: unct on t at s cont nuous ut not un orm y

continuous

Example

Letf(x) = 1/x, x(0, 2). Clearly the functionf is continuous on(0, 2). Assume thatf is uniformly continuous. Take = 1. Fix anarbitrary >0. Then, for x:= min(, 1) and y=x/2,

|x y|=|x/2|< (1)

and|f(x) f(y)|=|1/x 2/x|=|1/x| 1. (2)

Since (1) and (2) hold for any >0, we have that for = 1, thereexists no >0 such that|f(x) f(y)|< for all|x y|< .Therefore, f(x) = 1/x is not uniformly continuous.


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Definition

Let fbe a real valued function defined on an interval ofR.

Definition

For each x

[a, b], consider the function x: (a, b)

\ {x

} R,

x(t) =f(t) f(x)

t x , t(a, b), t=x.

Denote f(x) := limtxx(t). Iff(x) exists, then the function f

is said to be differentiableatx, andf

(x) is called the derivativeoff at x.


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Iff(x) is differentiable at every point of a set E, we say thatf isdifferentiableon E.

Similar to left and right hand limits, we can consider left and righthand derivatives at a point x as the left and right hand limits at apoint x of the function x(t).

Iff is defined on [a, b], the derivative off(x) at x= a is the righthand derivative and the derivative off(x) at x= b is the left handderivative provided they exist.


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Chain rule of differentiation

Theorem

Supposef is continuous on [a, b] and is differentiable at somepointx[a, b], andg is defined on an interval which contains therange offand is differentiable at the pointf(x). If

h(t) =g(f(t)), t[a, b],

then h is differentiable atx, andh(x) =g(f(x))f(x).

Find the derivative of the following functions: f(x) =c,

f(x) =xn, f(x) =x sin( 1x

)I{x= 0}, f(x) =x2 sin( 1x

)I{x= 0}.


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Local Maximum and Local Minimum

DefinitionLet fbe a real function defined on a metric space X. Thefunction f is said to have a local maximum at a point pX, if

>0 such that f(q)

f(p)

q

X withd(p, q)< . (3)

The function f is said to have a local minimumat a point pX,if (3) holds with the inequalityreplaced by.

Theorem

Letfbe defined on[a, b]. Iffhas a local extremum at a pointx(a, b) and iff(x) exists, then f(x) = 0.


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Examples

Example (Derivative can be zero at a point which is not a local

extremum)

f(x) =x3. Then f(x) = 3x2 exists for all xR. Atx= 0,f(x) = 0. However, f(x)> 0 for all xR \ {0}. Hence, x= 0is neither a point of local maximum nor a point of local minimum.

Example (Derivative may not exist at a point of local maximum)

f(x) =

x x


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Mean value theorem

TheoremIff andg are continuous real functions on [a, b] which aredifferentiable in (a, b), then there is a pointx(a, b) such that

(f(b)

f(a))g(x) = (g(b)

g(a))f(x).

Corollary

Iff is a continuous real function on [a, b] which is differentiable in(a, b), then there is a pointx(a, b) such that

f(b) f(a) = (b a)f(x).


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Applications

1. Suppose you drive a car from toll booth on a toll road toanother toll booth at an average speed of70 miles per hour. Whatcan be concluded about your actual speed during the trip? Inparticular, did you exceed the 65 mile per hour speed limit?


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Applications

1. Suppose you drive a car from toll booth on a toll road toanother toll booth at an average speed of70 miles per hour. Whatcan be concluded about your actual speed during the trip? Inparticular, did you exceed the 65 mile per hour speed limit? Yes

A li i


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Applications

1. Suppose you drive a car from toll booth on a toll road toanother toll booth at an average speed of70 miles per hour. Whatcan be concluded about your actual speed during the trip? In

particular, did you exceed the 65 mile per hour speed limit? Yes

2. Suppose two different functions have the same derivative. What

can you say about the relationship between the two functions?

A li i


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Applications

1. Suppose you drive a car from toll booth on a toll road toanother toll booth at an average speed of70 miles per hour. Whatcan be concluded about your actual speed during the trip? Inparticular, did you exceed the 65 mile per hour speed limit? Yes

2. Suppose two different functions have the same derivative. Whatcan you say about the relationship between the two functions?

Theorem

Iff(x) =g(x) for everyx(a, b), then for some constantk,f(x) =g(x) +k for allx(a, b).


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Theorem

Supposef is differentiable in (a, b).

(a) Iff

(x)0 for allx(a, b), then f is monotonicallyincreasing.(b) Iff(x) = 0 for allx(a, b), then f is constant.(c) Iff(x)0 for allx(a, b), then f is monotonicallydecreasing.

Fi t d i ti t t


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First derivative test

Theorem

Letf : (a, b)R. Letc(a, b) such thatf is continuous atcand differentiable on some open interval containingc, exceptpossibly atc itself.

(a) If there exists >0 such thatf

(x)0 for allx(c , c)andf(x)0 for allx(c, c + ), thenfhas a local maximum atc.(b) If there exists >0 such thatf(x)0 for allx(c , c)andf(x)

0 for allx

(c, c+), then fhas a local minimum at

c.(c) If there exists >0 such thatf(x)0 or iff(x)0 for allfor allx((c , c + ) \ {c}), thenfhas no local extremum atc.

I t di t l th


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Intermediate value theorem

Theorem

Letfbe a continuous real function on the interval [a, b]. Iff(a)< f(b) and ifc is a number such thatf(a)< c < f(b), thenthere exists a pointx(a, b) such thatf(x) =c.

Example (Continuity is not required to assume intermediate values)

Consider the function f satisfyingf(x) =sin(1/x) for all x >0and f(0) = 0. Then, f is discontinuous at x= 0. However, it hasthe intermediate value property.

Theorem

Supposef is a real differentiable function on [a, b] and supposef(a)< < f(b). Then there is a pointx(a, b) such thatf(x) =.

Take away: A function need not be continuous to assumeintermediate values.

Classification of discontinuities of a function


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Classification of discontinuities of a function

Theorem

Letfbe a monotonic function on (a, b). Then fhas nodiscontinuities of the second kind, and the set of points of(a, b) atwhich f is discontinuous is at most countable.

Theorem

Iff is differentiable on [a, b], then f cannot have any simplediscontinuities on [a, b].

Example (Function can be differentiable everywhere but thederivative need not be continuous)

Let f(x) =x2 sin(1/x)I{x= 0}. Then

f(x) =

2x sin(1/x) cos(1/x) x= 0,

0 x= 0.

Evaluation of limits


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Evaluation of limits

Theorem (LHospitals rule)

Supposef andg are real valued differentiable functions defined on(a, b) R, g(x)= 0 for allx(a, b), and for somec(a, b) andL

Rf(x)g(x)

L as xc.

If limxc f(x) = limxcg(x) = 0 or are both+, then

f(x)g(x)L as xc.

Taylors theorem


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Taylor s theorem

TheoremSupposef is a real function on [a, b], n is a positive integer,f(n1) is continuous on [a, b], f(n) exists for everyt(a, b). Let, be distinct points of [a, b], and define

P(t) =n1k=0

f(k)()

k! (t )k.

Then there exists a pointx between and such that

f() =P() +f(n)(x)

n! ( )n.

Real Analysis Lecture ppt

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