Determine if f defined by`f(x)={x^2sin1/x , ifx!=0 0, ifx=0`

Explanation

It is evident that `f` is defined at all points of the real line.


Let `c` be a real number.


Case I:


If `c neq 0`, then `f(c)=c^{2} sin (frac{1}{c})`



`lim _{x -> c} f(x) lim _{x -> c}(x^{2} sin frac{1}{x})=(lim _{x -> c} x^{2})(lim _{x -> c} frac{sin 1}{x})=c^{2} sin (frac{1}{c}) `


`therefore lim _{x -> c} f(x)=f(c)`



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