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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?
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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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