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**Continuity & Differentiability **

**Question 1: Prove that the function **

**Answer**

Therefore, f is continuous at x = 5

**Question 2: Examine the continuity of the function **

**Answer**

Thus, f is continuous at x = 3

**Question 3: Examine the following functions for continuity. (a) **

**(b)****(c) **

**(d) **

**Answer**

(c) The given function is

For any real number c â‰ âˆ’5, we obtain

Therefore, f is continuous at all real numbers greater than 5.

Hence, f is continuous at every real number and therefore, it is a continuous function.

**Question 4: Prove that the function **

The given function is f (x) = x

It is evident that f is defined at all positive integers, n, and its value at n is n

Therefore, f is continuous at n, where n is a positive integer.

**Question 5: Is the function f defined by**

**continuous at x = 0? At x = 1? At x = 2?****Answer**

The given function f is

At x = 0,

It is evident that f is defined at 0 and its value at 0 is 0.

The right hand limit of f at x = 1 is,

Therefore, f is continuous at x = 2

**Question 6: Find all points of discontinuity of f, where f is defined by**

**Answer**

The given function f is

It is evident that the given function f is defined at all the points of the real line.

Let c be a point on the real line. Then, three cases arise.

(i) c < 2

(ii) c > 2

(iii) c = 2

Case (i) c < 2

Therefore, f is continuous at all points x, such that x < 2

Case (ii) c > 2

It is observed that the left and right hand limit of f at x = 2 do not coincide.

Therefore, f is not continuous at x = 2

Hence, x = 2 is the only point of discontinuity of f.

**Question 7: Find all points of discontinuity of f, where f is defined by**

**Answer**

The given function f is

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x < âˆ’3

Case II:

It is observed that the left and right hand limit of f at x = 3 do not coincide.

Therefore, f is not continuous at x = 3

Case V:

Therefore, f is continuous at all points x, such that x > 3

Hence, x = 3 is the only point of discontinuity of f.

**Question 8: Find all points of discontinuity of f, where f is defined by****Answer**

The given function f is

It is observed that the left and right hand limit of f at x = 0 do not coincide.

Therefore, f is not continuous at x = 0

Case III:

Therefore, f is continuous at all points x, such that x > 0

Hence, x = 0 is the only point of discontinuity of f.

**Question 9: Find all points of discontinuity of f, where f is defined by**

**Answer**

The given function f is

Therefore, the given function is a continuous function.

Hence, the given function has no point of discontinuity.**Question 10: Find all points of discontinuity of f, where f is defined by**

**Answer**

The given function f is

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x > 1

Hence, the given function f has no point of discontinuity.**Question 11: Find all points of discontinuity of f, where f is defined by**

**Answer**

The given function f is

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x > 2

Thus, the given function f is continuous at every point on the real line.

Hence, f has no point of discontinuity.**Question 12: Find all points of discontinuity of f, where f is defined by**

**Answer**

The given function f is

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x > 1

Thus, from the above observation, it can be concluded that x = 1 is the only point of discontinuity of f.**Question 13: Is the function defined by**

**a continuous function?****Answer**

The given function is

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points x, such that x > 1

Thus, from the above observation, it can be concluded that x = 1 is the only point of

discontinuity of f.

**Question 14: Discuss the continuity of the function f, where f is defined by**

**Answer**

The given function is

The given function is defined at all points of the interval [0, 10].

Let c be a point in the interval [0, 10].

Case I:

Therefore, f is continuous at all points of the interval (1, 3).

Case IV:

Therefore, f is continuous at all points of the interval (3, 10].

Hence, f is not continuous at x = 1 and x = 3

**Question 15: Discuss the continuity of the function f, where f is defined by**

**Answer**

The given function is

The given function is defined at all points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at all points of the interval (0, 1).

Case IV:

Therefore, f is continuous at all points x, such that x > 1

Hence, f is not continuous only at x = 1

**Question 16: Discuss the continuity of the function f, where f is defined by**

**Answer**

The given function f is

The given function is defined at all points of the real line.

Let c be a point on the real line.

Case I:

Therefore, f is continuous at x = âˆ’1

Case III:

Therefore, f is continuous at all points x, such that x > 1

Thus, from the above observations, it can be concluded that f is continuous at all points of the real line.

**Question 17: Find the relationship between a and b so that the function f defined by is continuous at x = 3.**

**Answer**

The given function f is

If f is continuous at x = 3, then

**Question 18: For what value of is the function defined by**

**continuous at x = 0? What about continuity at x = 1?****Answer**

The given function f is

If f is continuous at x = 0, then

Therefore, there is no value of Î» for which f is continuous at x = 0 At x = 1,

f (1) = 4x + 1 = 4 Ã— 1 + 1 = 5

Therefore, for any values of Î», f is continuous at x = 1

**Question 19: Show that the function defined by g(x)= x-[x] is discontinuous at all integral point. Here [x] denotes the greatest integer less than or equal to x.**

The given function is

It is evident that g is defined at all integral points.

Let n be an integer.

Then,

It is observed that the left and right hand limits of f at x = n do not coincide.

Therefore, f is not continuous at x = n

Hence, g is discontinuous at all integral points.

**Question 20: Is the function defined by **

The given function is

It is evident that f is defined at x = p

Therefore, the given function f is continuous at x = Ï€

**Question 21: Discuss the continuity of the following functions. (a) f (x) = sin x + cos x (b) f (x) = sin x âˆ’ cos x (c) f (x) = sin x Ã— cos x**

It is known that if g and h are two continuous functions, then

It has to proved first that g (x) = sin x and h (x) = cos x are continuous functions.

Let g (x) = sin x

It is evident that g (x) = sin x is defined for every real number.

Let c be a real number. Put x = c + h

If x â†’ c, then h â†’ 0

Therefore, g is a continuous function.

Let h (x) = cos x

It is evident that h (x) = cos x is defined for every real number.

Let c be a real number. Put x = c + h

If x â†’ c, then h â†’ 0

h (c) = cos c

Therefore, h is a continuous function.

Therefore, it can be concluded that

(a) f (x) = g (x) + h (x) = sin x + cos x is a continuous function

(b) f (x) = g (x) âˆ’ h (x) = sin x âˆ’ cos x is a continuous function

(c) f (x) = g (x) Ã— h (x) = sin x Ã— cos x is a continuous function.

**Question 22: Discuss the continuity of the cosine, cosecant, secant and cotangent functions,Answer:**

It is known that if g and h are two continuous functions, then

It has to be proved first that g (x) = sin x and h (x) = cos x are continuous functions. Let g (x) = sin x

It is evident that g (x) = sin x is defined for every real number.

Let c be a real number. Put x = c + h

Therefore, g is a continuous function.

Let h (x) = cos x

It is evident that h (x) = cos x is defined for every real number.

Let c be a real number.

Put x = c + h

If x â†’ c, then h â†’ 0

Therefore, h (x) = cos x is continuous function.

Therefore, cotangent is continuous except at x = np, n ÃŽ Z

**Question 23: Find the points of discontinuity of f, where **

**Answer**

Therefore, f is continuous at x = 0

From the above observations, it can be concluded that f is continuous at all points of

the real line.

Thus, f has no point of discontinuity.**Question 24:Determine if f defined by**

**is a continuous function?****Answer**

The given function f is

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:

Therefore, f is continuous at x = 0

From the above observations, it can be concluded that f is continuous at every point of the real line.

Thus, f is a continuous function.

**Question 25: Examine the continuity of f, where f is defined by**

**Answer**

The given function f is

It is evident that f is defined at all points of the real line.

Let c be a real number.

Case I:

Therefore, f is continuous at x = 0

From the above observations, it can be concluded that f is continuous at every point of

the real line.

Thus, f is a continuous function.

**Question 26: Find the values of k so that the function f is continuous at the indicated point.**

**Answer**

Therefore, the required value of k is 6.

**Question 27: Find the values of k so that the function f is continuous at the indicated point.**

The given function is

The given function f is continuous at x = 2, if f is defined at x = 2 and if the value of f at x = 2 equals the limit of f at x = 2

It is evident that f is defined at x = 2 and f(2) = k(2)^{2} = 4k

**Question 28: Find the values of k so that the function f is continuous at the indicated point.**

**Answer**

The given function is

The given function f is continuous at x = p, if f is defined at x = p and if the value of f at

x = p equals the limit of f at x = p**Question 29: Find the values of k so that the function f is continuous at the indicated point.**

**Answer**

The given function f is

The given function f is continuous at x = 5, if f is defined at x = 5 and if the value of f at

x = 5 equals the limit of f at x = 5

**Question 30: Find the values of a and b such that the function defined by**

, **is a continuous function.****Answer**

The given function f is

It is evident that the given function f is defined at all points of the real line.

If f is a continuous function, then f is continuous at all real numbers.

In particular, f is continuous at x = 2 and x = 10

Since f is continuous at x = 2, we obtain

Therefore, the values of a and b for which f is a continuous function are 2 and 1 respectively.

**Question 31: Show that the function defined by f(x) = cos (x ^{2}) is a continuous function.**

The given function is f (x) = cos (x

This function f is defined for every real number and f can be written as the composition

of two functions as,

f = g o h, where g (x) = cos x and h (x) = x

Therefore, g (x) = cos x is continuous function.

h (x) = x

Clearly, h is defined for every real number.

Let k be a real number, then h (k) = k

Therefore, h is a continuous function.

It is known that for real valued functions g and h, such that (g o h) is defined at c, if g is continuous at c and if f is continuous at g (c), then (f o g) is continuous at c.

Therefore, is a continuous function.

**Question 32: Show that the function defined by** **is a continuous function.****Answer**

The given function is

This function f is defined for every real number and f can be written as the composition of two functions as,

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:

Therefore, g is continuous at all points x, such that x > 0.

**Question 33: Examine that sin|x| is a continuous function.**

**Answer**

This function f is defined for every real number and f can be written as the composition

of two functions as,

Clearly, g is defined for all real numbers.

Let c be a real number.

Case I:

Therefore, g is continuous at x = 0

From the above three observations, it can be concluded that g is continuous at all

points. h (x) = sin x

It is evident that h (x) = sin x is defined for every real number.

Let c be a real number. Put x = c +k

If x â†’ c, then k â†’ 0 h (c) = sin c

Therefore, h is a continuous function.

It is known that for real valued functions g and h,such that (g o h) is defined at c, if g is continuous at c and if f is continuous at g (c), then (f o g) is continuous at c.

is a continuous function.

**Question 34:Find all the points of discontinuity of f defined by **

**Answer**

The given function is

The two functions, g and h, are defined as

Clearly, g is defined for all real numbers.

Let c be a real number.

**Case I:**

From the above three observations, it can be concluded that g is continuous at all

**Case II: **

Clearly, h is defined for every real number.

Let c be a real number.

Case I:

Therefore, h is continuous at all points x, such that x < âˆ’1

Case II:

Therefore, h is continuous at x = âˆ’1

From the above three observations, it can be concluded that h is continuous at all points of the real line. g and h are continuous functions. Therefore, f = g âˆ’ h is also acontinuous function.

Therefore, f has no point of discontinuity.

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