Analysis period 1.4

Question 1

Local continuity

Answer 1

Suppose that a function f : D-> R with D c- R. Then f is continuous at a ∈ D if and only if for all eps. > 0 there exist delta > 0 |f(x) - f(a)| < eps. for all x ∈ D s.t. |x - a| < delta so lim x->a f(x) = f(a)

Question 2

Global continuity

Answer 2

f ; D -> R is continuos on a set E c_ if and only if f is continuous at each point x∈E. If f is continuous at every point in its domain D, we simpply say that f is continuous.

Question 3

Proving f has limit L∈R when x tend to a∈R

Answer 3

1. Let eps.>0 be given
2. Start with |f(x) - f(a)|
3. replace |f(x) - f(a)| to c . |x - a| with constant c, so you can replace |x - a| with delta
4. Next, if you solve some expression of delta for epsilon, you can choose your delta
5. State that there exist delta > 0 s.t. |f(x) -L| < eps. provided that |x - a| < delta and draw your conlusion

Question 4

Side continuity

Answer 4

Suppose that a function f ; D-> R with D c_R then f is right continuous at a , meaning that f is continuous from the right at a, if and only if for all eps.>0 there exists delta>0 |f(x) - f(a)| < eps provided that 0<= x - a < delta and x∈D

Suppose that a function f ; D-> R with D c_R then f is left continuous at a , if and only if for all eps.>0 there exists delta>0 |f(x) - f(a)| < eps provided that 0<= x - a < delta and x∈D

Question 5

continuity of bounded functions

Answer 5

Suppose that D is the domain of f
1. if f is continuous at a, f is bounded on the set ( a - delta, a + delta )
2. if f is right continuous at a , f is bounded on set [a, a + delta)
3. if f is left continous at a, f is bounded on set (a - delta, a]
4. if f is continuous at a and f(a)>0 then there exist delta>0 s.t f(x)>0.5f(a) for all x is an element of (a-delta, a + delta)

Question 6

Properties continuous functions

Answer 6

suppose that functions f,g: D->R with Dc_R are continuous at a then
1. f +- g are continuous at a
2. fg are continuous at a
3. f/g are continuous at a, provided that g(a) =/ 0

Question 7

Continuity of composite functions

Answer 7

consider functions f : A->R and g : B->R with A,Bc_R s.t. f(a)c_B. If f is continuous at some x=a is an element of A and g is continuous at some b = f(a) is an element of B,, then the function g * f is continuous at x=a

Question 8

removable discontinuity

Answer 8

a function f: D->R with Dc_R has a removable discontinuity at x=a if either
1. a is not an element of D and lim x->a f(x) is finite
2. a is not an element of D and lim x->a f(x) = L is not an element of f(a)

Question 9

continuous extension

Answer 9

if g is continuous, then g is called a continuous extension of f. Thus if g extends f , all values of g agree with values of f and g has some additional values.

Question 10

Jump discontinuity

Answer 10

suppose further that L=/M. then f has a jump discontinuity at x=a. the value j(f) = L - M is called the jump of f at x = a.

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Question 11

Infinite discontinuity

Answer 11

one of the following three conditions is satisfied
1. lim x->a+ and limx->a are both infinite
2. a is not an accumulation point and lim x->a- f(x) is infinite
3. a is not an accumulation point and lim x->a+ f(x) is infinite
then f has an infinite discontinuity at x=a

Question 12

Piecewise discontinuity

Answer 12

function f is piecewise continuous if and only if there exists finitely many points x1,x2,……,xn:
1. f is continuous on D expect at x1,x2,x3…..,xn and
2. f has simple discontinuities at x1,x2,x3,……,xn