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# Best Homework Help For Continuity Of Functions

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## Continuity Of Functions:

A function which does not have a break in its graph is called a continuous function. A function f(x) is said to be continuous at a point x=a, if f(a) exists and the value of f(a) is finite, left hand limit is equal to the right hand limit and both are finite and f(x) =  f(a)

## Continuity Of Functions Sample Questions:

Question 1: Let S ={ f: RR | > 0 such that ∀δ> 0, |x-y|< δ, |f(x)- f(y)|< } then

Explanation: Check the option (a) and (b) they are incorrect as the function is not continuous.

Question 2: Let f:(0, )  R be uniformly continuous  then

(a) f(x) and f(x) exist

(b) f(x) need not exist but f(x) exists

(c) f(x) exists and f(x) need not exist

(d) None exists

Explanation: A function is uniformly continuous if  for every  > 0 there exist.

Question 3: Which of the function is uniform and continuous on interval (0,1)

Explanation: f(x) is continuous in (a,b) and if limits exist at endpoints then f(x) is uniformly continuous.

Question 4: f: RR is such that f(0) = 0 and |f’(x)|  5 for all x, we can conclude that f(1) is in

(a) (5,6)

(b) [-5,5]

(c) [1,5]

(d) [-4,4]

Explanation: |f’(x)|  5  =>  -5  f’(x)  5
f’(x) - 5  0
f’(x) – 5 = a

Question 5: Which of the following is uniform continuous on (0,1).

Explanation:

Question 6: Let f: RR be a continuous and one-one function then which of the following is true

(a) f is onto

(b) f is either strictly increasing or strictly decreasing

(c) there exists x  R

(d) f is unbounded

Explanation: f: RR is continuous and one-one.

Question 7:

Explanation: It is a sequence of continuous function at x= 0

Question 8: f(x) = log (x)

(a) f(x) is discontinuous on R

(b) f(x) is continuous on R

(c) f(x) is continuous on (0, ∞)

(d) f(x) is defined on R

Explanation: log(x) is continuous on (0, ∞), has domain R

Question 9: Which of the following statements is incorrect.

(a)|x| is continuous

(b) tan x cot x and sec x are continuous in their respective domains

(c) Addition and subtraction of a discontinuous function with continuous function may be continuous or discontinuous.

(d) Product  and ratio of two continuous function is always continuous provided the denominator is non-zero at a point where we are checking the continuity.

Explanation: Addition and subtraction of a discontinuous function.

Question 10: f(x) = |6-x +|x|| is

(a) Discontinuous on R

(b) Continuous on R

(c) Discontinuous at x = 1

(d) has a removable type of discontinuity