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Question
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
`f (x) = {(x, if x<=1),(5, if x > 1):}`
(i) ⇒ x = 0 पर
`lim_(x -> 0^-) f (x) = lim_(h -> 0) f(0 - h)`
= 0 - 0
= 0
`lim_(x -> 0^+) f(x) = lim_(h -> 0) f(0 + h)`
= 0 + 0
= 0
f(0) = 0
(ii) Hence, f is continuous at x = 0.
⇒ at x = 1
`lim_(x -> 1)` f(x) = `lim_(h -> 1)` f (1 - h)
= 1 - 0
= 1
`lim_(x -> 1)` f(x) = 5
Hence f is not continuous at x = 1.
⇒ at x = 2
`lim_(x -> 2^-)` f(x) = `lim_(h -> 2^+)` f(x) = 5
f(2) = 5
Hence, f is continuous at x = 2.
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