let f(x) = x^(2) sin (1/x^2) for 0<x< or = 1 and f(0)=0. prove that f is continuous on [0,1].

the answer said that since lx^(2) sin (1/x^2)l < or = l x^2 l, then f is continuous on 0 and hence continuous on [0,1]..but i do not see how this statement proves the question.

i thought to prove continuous, it would be assume that f is cont at c where c is between [0,1]

then

l x^(2) sin (1/x^2) - c^(2) sin (1/c^2)l < l x^2 - c^2l < 2 l x-cl

thus continuous?