Proof space is Connected Space

Question

Let $A:=\{(0,y):y\in[-1,1]\}$ and $B:=\{(x, sin({1\over x}):x \in (0,1]\}$, finally, let $C:=A\cup B$

1. Show C is a connected space
2. Show C is not path-connected

1 has me stumped, I have only ever proven that a space is not connected.

2 Is duplicate, look here [proof that topologist sine curve is not locally connected