Path connectedness of a subset of $\mathbb{R}^2$

Question

I know about Connectedness and Path Connectedness in Topology. Recently, I got stuck into a problem. Let $A$ be the following subset of $\mathbb{R}^2:$ $$A=\{(x,y) \mid (x+1)^2 + y^2 \leq 1\}\cup \{(x,y) \mid y=x\sin(\frac{1} {x}),\ x>0\}$$ The connectedness is obvious from if $A$ is connected then $cl(A)$ is also connected. But I can't prove the path connectedness ( The answer is given that $A$ is both connected & Path connected). Please help. Thanks in advance.

Answer 1

Let $B$ be the second set. It should be clear that $B$ is path connected.

Since $A$ is convex, it is path connected.

Let $p(x) = \begin{cases} (x,x \sin { 1\over x}) , & x \in (0,1]\\ (0,0), & x = 0 \end{cases}$.

Note that $p$ is continuous, $p(0) \in A$ and $p(1) \in B$.

Hence any two points of $A\cup B$ can be connected by a path.