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Question
Show that the function defined by f (x) = cos (x2) is a continuous function.
Solution
Let f (x) = cos (x2).
Domian of f = R.
Let a be any arbitrary real number.
`lim_(x->a^-) f (x) = lim_(h->0) cos (a - h)^2 = cos a^2`
`lim_(x->a^+) f (x) = lim_(h->0) cos (a + h)^2 = cos a^2`
Also f (a) = cos a2
Thus, `lim_(x->a^-) f (x) = lim_(x->a^+) f (x) = f (a) AA a ∈ R`
∴ f (x) = cos(x2) is continuous at a ∀ a ∈ R.
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