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Question
Show that the function f defined by f(x) = `{{:(x sin 1/x",", x ≠ 0),(0",", x = 0):}` is continuous at x = 0.
Solution
Left hand limit at x = 0 is given by
`lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) x sin 1/x` = 0 ....`["since", -1 < sin 1/x < 1]`
Similarly `lim_(x -> 0^+) "f"(x) = lim_(x -> 0^+) x sin 1/x` = 0.
. Moreover f(0) = 0.
Thus `lim_(x -> 0^-) "f"(x) = lim_(x -> 0^-) "f"(x)`
= f(0)
Hence f is continuous at x = 0.
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