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Question
Discuss the continuity of the function f, where f is defined by `f(x) = {(2x , ","if x < 0),(0, "," if 0 <= x <= 1),(4x, "," if x > 1):}`
Solution
`"f"("x") = {(2"x""," " if", "x" < 0),(0"," " if", 0 le "x" le 1),(4"x" "," " if", "x" > 1):}`
For x < 0, f(x) = 2x;
0 < x < 1, f(x) = 0 and
for x > 1, f(x) = 4 x is a polynomial and continuous function.
So this is a function.
At x = 0,
`lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-)` (2 x)
= `lim_("h" -> 0)` [2 (0 - h)]
= `lim_("h" -> 0)` (-2h)
= - 2 × 0
= 0
`lim_(x -> 0^+) "f"(x) = lim_(x -> 0^+)` (0) = 0
Hence, f is continuous at x = 0.
At x = 1,
`lim_(x -> 1^-) "f"(x) = lim_(x -> 1^-)` (0) = 0
`lim_(x -> 1^+) "f"(x) = lim_(x -> 1^+)` (4x)
= `lim_("h" -> 0)` [4 (1 + h)]
= `lim_("h" -> 0)` (4 + 4h)
= 4 + 4 × 0
= 4
Hence, it is not continuous at x = 1.
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