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