Question

Determine if f defined by

f (x) = x² sin 1/x,x≠ 0

f(x)=0,     if x =0

Answer 1

 The given function f is  

It is evident that f is defined at all points of the real line. Let c be a real number.

Case I:

Therefore, f is continuous at all points x ≠ 0

Case II:

Therefore, f is continuous at x = 0
From the above observations, it can be concluded that f is continuous at every point of the real line. Thus, f is a continuous function.