Question

Determine if f defined by f(x) = `{(x^2 sin  1/x", if"  x != 0),(0", if"  x = 0):}` is a continuous function?

Answer 1

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.