Class 12 Maths Continuity Differentiability Introduction to Continuity

Introduction to Continuity

The meaning of the term continuity is same as we use in our daily life. For example the water flow in the rivers is continuous. The flow of time in human life is continuous i.e. we are getting older continuously and so on. Similarly, in mathematics, we have the notion of the continuity of a function.

When we say that a function f(x) is continuous at a point x = a it means that the point (a, f(a)), the graph of the function has no holes or gaps. In simple words, a function is said to be continuous if we can sketch its curve on a graph without lifting your pen even once.

Definition 1. Suppose f is a real function on a subset of the real numbers and let c be a point in the domain of f. Then f is continuous at c if limx-> c f(x) = f(c)

Problem: Prove that the function f(x) = 5x – 3 is continuous at x = 0, x = -3 and x = 5

Solution:

The given function is f(x) = 5x – 3

At x = 0, f(0) = 5 * 0 – 3 = -3

limx->0 f(x) = limx->0 (5x - 3) = 5 * 0 – 3 = -3

Therefore, f is continuous at x = 0

At x = -3, f(-3) = 5 * (-3) – 3 = -15 – 3 = -18

limx->-3 f(x) = limx->-3 (5x - 3) = 5 * (-3) – 3 = -15 – 3 = -18

Therefore, f is continuous at x = −3

At x = 5, f(5) = 5 * 5 – 3 = 25 – 3 = 22

limx->5 f(x) = limx->5 (5x - 3) = 5 * 5 – 3 = 25 – 3 = 22

Therefore, f is continuous at x = 5

Definition 2. If the left hand limit, right hand limit and the value of the function at x = c exist and equal to each other, then f is said to be continuous at x = c. i.e.

limx-> c- f(x) = f(c) = limx-> c+ f(x) = f(c)

Again if the right hand and left hand limits at x = c coincide, then we say that the common value is the limit of the function at x = c. So, we may also represent the definition of continuity as follows:

Definition 3. A function is continuous at x = c if the function is defined at x = c and if the value of the function at x = c equals the limit of the function at x = c. If f is not continuous at c, we say f is not continuous at c and c is called a point of discontinuity of f.

Problem: Is the function f defined by

f(x) =    x, if x ≤ 1

             5, if x > 1

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

Solution:

The given function f is

f(x) =    x, if x ≤ 1

             5, if x > 1

At x = 0,

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

limx->0 f(x) = limx->0 x = 0

So, limx->0 f(x) = f(0)

Therefore, f is continuous at x = 0

At x = 1, f is defined at 1 and its value at 1 is 1.

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

limx->1- f(x) = limx->1- x = 1

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

limx->1+ f(x) = limx->1+ (5) = 5

So, limx->1- f(x) = limx->1+ f(x)

Therefore, f is not continuous at x = 1

At x = 2, f is defined at 2 and its value at 2 is 5.

limx->2 f(x) = limx->2 (5) = 5

So, limx->2 f(x) = f(2)

Therefore, f is continuous at x = 2

Definition 4. Definition 2 A real function f is said to be continuous if it is continuous at every point in the domain of f.

Let f is a function defined on a closed interval [a, b], then for f to be continuous, it needs to be continuous at every point in [a, b] including the end points a and b.

Problem: Find all points of discontinuity of f, where f is defined by

f(x) =    x3 - 3, if x ≤ 2

             x2 + 1, if x > 2

Solution:

The given function f is

f(x) =    x3 - 3, if x ≤ 2

             x2 + 1, if x > 2

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:

If c < 2, then f(c) = c3 - 3

And limx->c f(x) = limx->c (x3 - 3) = c3 - 3

So, limx->c f(x) = f(c)

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

Case II:

If c = 2, then f(c) = f(2) = 23 - 3 = 8 – 3 = 5

The left hand limit of f at x = 2 is,

limx->2- f(x) = limx->2- (x3 - 3) = 23 - 3 = 8 – 3 = 5

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

limx->2+ f(x) = limx->2+ (x2 + 1) = 22 + 1 = 4 + 1 = 5

Therefore, f is continuous at x = 2

Case III:

If c > 2, then f(c) = c2 + 1

limx->c f(x) = limx->c (x2 + 1) = c2 + 1

So, limx->c f(x) = f(c)

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.

Algebra of continuous functions:

Let f and g be two real functions continuous at a real number c. Then

  1. f + g is continuous at x = c.
  2. f - g is continuous at x = c.
  3. f * g is continuous at x = c.
  4. f/g is continuous at x = c, where g(c) ≠ 0.

Problem: Discuss the continuity of the functions:

(a) f(x) = sin x + cos x  (b) f (x) = sin x - cos x  (c) f (x) = sin x * cos x

Solution:

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

g + h, g – h and g * h are also continuous.

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 -> c, then h -> 0

g(c) = sin c

limx->c g(x) = limx->c sin x

                   = limh->0 sin (c + h)

                   = limh->0 [sin c * cos h + cos c * sin h]

                   = limh->0 [sin c * cos h] + limh->0 [cos c * sin h]

                   = sin c * cos 0 + cos c * sin 0

                   = sin c * 1 + cos c * 0

                   = sin c + 0

                   = sin c

So, limx->c g(x) = g(c)

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 ->c, then h >0

h (c) = cos c

limx->c h(x) = limx->c cos x

                   = limh->0 cos (c + h)

                   = limh->0 [cos c * cos h - sin c * sin h]

                   = limh->0 [cos c * cos h] - limh->0 [sin c * sin h]

                   = cos c * cos 0 + sin c * sin 0

                   = cos c * 1 + sin c * 0

                   = cos c + 0

                   = cos c

So, limx->c h(x) = h(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) − h (x) = sin x − cos x is a continuous function

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

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