# Is the function f defined by Continuous at x = 0? At x = 1? At x = 2?

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 < 1
f(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

As,

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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