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Question
Examine the continuity of f, where f is defined by `f(x) = {(sin x - cos x, if x != 0),(-1, "," if x = 0):}`
Solution
`"f"("x") = {("sin x" - "cos x""," " if" "x" ne 0),(-1"," " if" "x" = 0):}`
Approach1:
If f(x) is continuous at x = c, it implies:
f(c) `= lim_(x -> "c"^+) "f"(x) = lim_(x -> "c"^-) "f"(x)`
`=> -1 = sin 0 - cos 0 = -sin 0 - cos 0`
`=> -1 = -1 = -1`
Which is true, i.e. f(x) is continuous at x = 0.
Approach2:
`c ne 0 and c sub R`
If f(x) is continuous at x = c, it implies:
sin c - cos c is continuous, i.e. sin c and cos c are continuous functions, which is true.
That is, f(x) is also continuous at `x ne 0`.
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