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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):}`
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Solution
f(x) = `{(2x", if" x < 0),(0", if" 0 <= x <= 1),(4x", if" x > 1):}`
For x < 0, f(x) = 2x;
0 < x < 1, f(x) = 0 and
for x > 1, f(x) = 4x is a polynomial and continuous function.
So this is a function.
At x = 0,
`lim_(x -> 0^-)` f(x) = `lim_(x -> 0^-)` (2x)
= `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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