The graph of that function is this:



Solve f(x) = 0
       2x² = 0
        x² = 0
         x = 0


f(0) = 3(0)² = 0

f(x) is continuous at x = 0 because

lim f(x) = lim 3x = 0 
x->0-      x->0-

lim f(x) = lim 2x² = 0 
x->0+      x->1+

Therefore

lim f(x) = lim f(x) = lim f(x) = f(0) = 0  
x->1-      x->1+      x->1

which proves that f(x) is continuous at x=0

However f(x) is not continuous at x=1 because

lim f(x) = lim 2x² = 3 
x->1-      x->1-

lim f(x) = lim x²-3 = -2 
x->1+      x->1+

Therefore 

lim f(x) ≠ lim f(x)  
x->1-      x->1+

Since f(x) is not continuous at x=1, it is not continuous 
everywhere on its domain.

Edwin