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Question
Discuss the continuity of the function f, where f is defined by `f(x) = {(3, ","if 0 <= x <= 1),(4, ","if 1 < x < 3),(5, ","if 3 <= x <= 10):}`
Solution
`f(x) = {(3, ","if 0 <= x <= 1),(4, ","if 1 < x < 3),(5, ","if 3 <= x <= 10):}`
For `0 le x le 1`, f(x) = 3;
1< x < 3, f(x) = 4 and
`3 < x le 10,` f(x) = 5, is a continuous function
So this is a function
At x = 1,
`lim_(x -> 1^-) f(x) = lim_(x -> 1^-)` (3) = 3
`lim_(x -> 1^+) f(x) = lim_(x -> 1^+)` (4) = 4
Hence, f is not continuous at x = 1.
At x = 3,
`lim_(x -> 3^-) f(x) = lim_(x -> 3^-)` (4) = 4
`lim_(x -> 3^+) f(x) = lim_(x -> 3^+)` (5) = 5
Hence, f is not continuous at x = 3.
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