Q. 54.0( 97 Votes )

# Is the function f defined by

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

Answer :

It is given that

__Case I:__ x = 0

We can see that f is defined at 0 and its value at 0 is 0.

LHL = RHL = f(0)

Hence, f is continuous at x = 0.

__Case II:__ x = 1

We can see that f is defined at 1 and its value at 1 is 1.

For x < 1f(x) = x

Hence, LHL:

= 1

For x > 1

f(x) = 5

therefore, RHL

= 5

Hence, f is not continuous at x = 1.

__Case III:__ x = 2

We can see that f is defined at 2 and its value at 2 is 5

LHL:

here f(2 - h) = 5, as h → 0 ⇒ 2 - h → 2

RHL:

LHL = RHL = f(2)

here f(2 + h) = 5, as h → 0 ⇒ 2 + h → 2

Hence, f is continuous at x = 2.

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