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Question
Determine if f defined by `f(x) = {(x^2 sin 1/x, "," if x != 0),(0, "," if x = 0):}` is a continuous function?
Solution
We have f (0) = 0
`lim_(x->0^-) f (x) = lim_(h->0)(0 - h^2) sin 1/-h = h^2 sin (1/h)`
but `sin 1/h ∈ [-1, 1]`
= `h^2 sin 1/h -> 0` as h ->0.
`lim_(x->0^+) f (x) = lim_(h->0) (0 + h)^2 sin 1/h =h^2 sin 1/h = 0 `
= `lim_(x->0^-) f (x) = lim_(x->0^+) f (x) = f (0)`
= f is continuous at x = 0
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