1. Determine whether or not the functions f & g are defined, and continuous at x

ID: 3011692 • Letter: 1

Question

1. Determine whether or not the functions f & g are defined, and continuous at x=0, x=1/2

f(x) = x if x=1/n with n in Z and 1-x if x is otherwise

2. Determine whether or not the functions f & g are defined, and continuous at x=0, x=1/2

f(x) = x if x is irrational and 1-x if x is irrational

3. Prove or find a counter example to the following statements, in which we assume that f and g are functions defined on the indicated intervals.

a) f bounded on [a,b] implies the f is continuous on [a,b]

b) f continuous on (a,b) implies that f is bounded on (a,b)

c) [f(x)]^2 continuous on (a,b) implies that f is continuous on (a,b)

d) f & g not continuous on (a,b) implies fg in not continous on (a,b)

e) f & g not continuous on (a,b) implies that f+g in not continuous on (a,b)

f) f & g not continuous on (a,b) implies that fog is not continuous on (a,b)

4. Give an example of a function f:[a,b] -> R that is not continuous but whose range is

a) an open and bounded interval

b) an open and unbounded interval

c) a closed and unbounded interval

5. Which is a stronger assumption on the function f, continuity or the intermediate value property? explain

6. Show that the equation 2^x = 3x has a solution x=c for some c in (0,1) using

a)Bolanzos intermediate value theorem

b) Brouwer's fixed-point theorem

note: only one question allowed per submission

1. Determine whether or not the functions f & g are defined, and continuous at x=0, x=1/2

f(x) = x if x=1/n with n in Z and 1-x if x is otherwise

a function is continuous if left hand limit = right hand limit = f(x)

so at x =0 f(x) = 1/0 = infinity , left hand limit at x ->0-h f(x) = 1/-h = - infinity

at rhl x-> 0+h f(x) = 1/0+h = +infinity

So Lhl not = rhl not = f(x) so limit does not exist.

at x =1/2 f(x) = 1-x

f(1/2 ) = 1-1/2 = 1/2

lhl at 1/2 = lt x->1/2-h = 1-( 1/2-h )= 1/2+h = 1/2

rhl at 1/2 = lt x->1/2+h = 1+( 1/2+h )= 3/2+h = 3/2

sine lhl is not equal to rhl , so does not exist

note here hends to 0

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