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Question
Examine the continuity of `"f"(x) {:(= sin x",", "for" x ≤ pi/4), (= cos x",", "for" x > pi/4):}} "at" x = pi/4`
Solution
`"f"(x) {:(= sin x";", x ≤ pi/4), (= cos x";", x > pi/4):}`
`lim_(x -> pi^-/4) "f"(x) = lim_(x -> pi^-/4) (sin x)`
= `sin pi/4`
= `1/sqrt(2)`
`lim_(x -> pi^+/4) "f"(x) = lim_(x -> pi^+/4) (cos x)`
= `cos pi/4`
= `1/sqrt(2)`
Also `"f"(pi/4) = sin pi/4`
= `1/sqrt(2)`
∴ `lim_(x -> pi^-/4) "f"(x) = lim_(x -> pi^+/4) "f"(x) = "f"(pi/4)`
∴ f(x) is continuous at x = `pi/4`
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