Determine continuity of f(x)={x2sin(1/x)x≠00x=0f(x) = \begin{cases} x^2 \sin(1/x) & x \neq 0 \\ 0 & x=0 \end{cases}f(x)={x2sin(1/x)0x=0x=0 at x=0x=0x=0.
Continuous, because limx→0x2sin(1/x)=0=f(0)\lim_{x \to 0} x^2 \sin(1/x) = 0 = f(0)limx→0x2sin(1/x)=0=f(0)
Discontinuous, because the limit doesn't exist
Continuous, because sin(1/x)\sin(1/x)sin(1/x) is bounded
Discontinuous, because f(0)f(0)f(0) is artificially defined