Is the function defined by f(x)= {(x + 5, if x ≤ 1) (x − 5, if x > 1) a continous function?

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)= {(x + 5, if x ≤ 1) (x − 5, if x > 1)

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

Let c be a point on the real line.

Case I:

If c < 1, then f(c) = c + 5

lim_x→c f(x) = lim_x→c (x + 5)

= c + 5

⇒ lim_x→c f(x) = f(c)

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

Case II:

If c = 1, then f(1) = 1 + 5 = 6

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

lim_x→1⁻ f(x) = lim_x→1⁻ (x + 5)

= 1 + 5 = 6

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

lim_x→1⁺ f(x) = lim_x→1⁺ (x − 5)

= 1 − 5

= −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 c > 1, then f(c) = c − 5

lim_x→c f(x) = lim_x→c (x − 5)

= c − 5

⇒ lim_x→c f(x) = f(c)

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

From the above observation, it can be concluded that x = 1 is the only point of discontinuity of f

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 13

Is the function defined by f(x)= {(x + 5, if x ≤ 1) (x − 5, if x > 1) a continous function?

Summary:

Function defined by f(x)= {(x + 5, if x ≤ 1) (x − 5, if x > 1) is not a continous function. x = 1 is the only point of discontinuity of f