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Question
Discuss the continuity of the cosine, cosecant, secant and cotangent functions,
Solution
(i) f (x) = cos (x)
Let c be any real number.
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x -> c^+) f(x) = lim_(x -> c^-) f (x)`
⇒ (cos c) = (cos c) = (cos c)
which is true, i.e. f(x) is continuous at every point on the real number line.
(ii) f(x) = cosec (x)
Let c be any real number.
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x ->^+) f(x) = lim_(x -> c^-) f(x)`
`=>` (cosec c) = (cosec c) = (cosec c)
which is true, i.e. f(x) is continuous at every point on the real number line.
(iii) f(x) = sec (x)
Let c be any real number.
If f(x) is continuous at x = c, this implies:
f(c) = `lim_(x -> c^+) f(x) = lim_(x -> c^-) f(x)`
`=>` (sec c) = (sec c) = (sec c)
which is true, i.e. f(x) is continuous at every point on the real number line.
(iv) f(x) = cot (x)
Let c be any real number such that (n - 1)`pi < x < npi,` where n represents an integer point.
If f(x) is continuous at x = c, this implies:
f(c) `= lim_(x -> c^+) f (x) = lim_(x -> c^-) f(x)`
`=>` (cot c) = (cot c) = (cot c)
Which is true, i.e. f(x) is continuous at every point on the real number line between (n - 1)`pi` and `n pi`.
Now if we consider c such that c = `n pi` where n represents an integer point, then:
If f(x) is continuous at x = c, this implies:
f(c) `= lim_(x -> c^+) f(x) = lim_(x -> c^-) f(x)`
`=> pm infty = pm infty = pm infty`
That is, f(x) is continuous at every point on the real number line except at the `n pi` type points.
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