How do you prove by definition that the function f ( x ) = x 2 sin ( 1 x ) is continuous at x=0?

Question

How do you prove by definition that the function             f              (        x        )              =          x      2              sin              (                  1          x                )             is continuous at x=0?

Leroy Cunningham · Accepted Answer

This function as is, is not continuous at             x      0        =    0   because it is not defined there.      D    f    =  {  x  ∈  R  :  x  ≠  0  }  =      R    ∗    =  (  −  ∞  ,  0  )  ∪  (  0  ,  +  ∞  )This function would be continuous for example,            f              (        x        )              =          {                                                  x              2                                      sin                              (                                  1                  x                                )                                                    ,                                       x            ≠            0                                                0                                         ,                                       x            =            0

How do you prove by definition that f(x)= x^2 sin (1/x) is continuous at x=0?

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