Question

Is the function y = ⎨x*sin(1/x), if x is not 0

continuous at x = 0 ? Solved Problem of this type.
x
⎪
0, if x = 0
⎩
Solution. A function f is continuous at a point a if lim f ( x ) = f ( a ) . Observe that for all
x→a
1
1
≤ 1. Hence − x ≤ x sin ≤ x , x ≠ 0. (Please make sure that you
x
x
understand why we needed to use the absolute value signs here!) Since
1
lim ( − x ) = lim x = 0, we conclude, by the Squeeze Theorem, that lim x sin = 0. This
x→0
x→0
x→0
x
means that the function is continuous at x = 0.

Answer 1

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