Discuss the continuity of the function f , where f is defined by f(x)= {(3, if 0 ≤ x ≤ 1) (4, if 1 < x < 3) (5, if 3 ≤ x ≤ 10)

Solution:

A function is said to be continuous when the graph of the function is a single unbroken curve.

The given function is 

f(x)= {(3, if 0 ≤ x ≤ 1) (4, if 1 < x < 3) (5, if 3 ≤ x ≤ 10)

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

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

Case I:

If 0 ≤ c < 1, then f(c) = 3

lim_x→c f(x) = lim_x→c (3)

= 3

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous in the interval [0,1)

Case II:

If c = 1, then f(3) = 3

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

lim_x→1⁻ f(x) = lim_x→1⁻ (3)

= 3

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

lim_x→1⁺ f(x) = lim_x→1⁺ (4)

= 4

It is observed that the left and right-hand limit of f at x = 1 do not coincide.

Therefore, f is not continuous at x = 1.

Case III:

If 1 < c < 3, then f(c) = 4

lim_x→c f(x) = lim_x→c (4) = 4

⇒  lim_x→c f(x) = f(c)

Therefore, f is continuous at in the interval (1,3).

Case IV:

If c = 3, then f(c) = 5

The left-hand limit of f at x=3 is,

lim_x→3⁻ f(x) = lim_x→3⁻ (4) = 4

The right-hand limit of f at x = 3 is,

lim_x→3⁺ f(x) = lim_x→3⁺ (5) = 5

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

Therefore, f is discontinuous at x = 3.

Case V:

If 3 < c ≤ 10, then f(c) = 5

lim_x→c f(x) = lim_x→c(5) = 5

⇒ lim_x→c f(x) = f(c)

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

Hence, f is discontinuous at x = 1 and x = 3

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 14

Discuss the continuity of the function f , where f is defined by f(x)= {(3, if 0 ≤ x ≤ 1) (4, if 1 < x < 3) (5, if 3 ≤ x ≤ 10)

Summary:

Function defined by f(x)= {(3, if 0 ≤ x ≤ 1) (4, if 1 < x < 3) (5, if 3 ≤ x ≤ 10), f is continuous at all points of the interval (3,10]. Hence, f is discontinuous at x = 1 and x = 3