I know about Connectedness and Path Connectedness in Topology. Recently, I got stuck into a problem. Let $A$ be the following subset of $\mathbb{R}^2:$ $$A=\{(x,y) \mid (x+1)^2 + y^2 \leq 1\}\cup \{(x,y) \mid y=x\sin(\frac{1} {x}),\ x&gt;0\}$$ 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 &amp; Path connected). Please help. Thanks in advance.
Let $A$ be the following subset of $\mathbb{R}^{2}$:$$A=\{(x,y):(x+1)^{2}+y^{2}\leq 1\}\cup\{(x,y):y=x\sin(\frac{1}{x}),x&gt;0\}$$ Then  $1.$ $A$ is connected.  $2.$ $A$ is compact.  $3.$ $A$ is path connected.  $4.$ $A$ is bounded.  It is clear that $A$ is unbounded as the graph of $x\sin(\frac{1}{x})$, so option $2$nd and $4$th are not correct. Again $A$ is connected by using the fact that closure of connected set is connected. But i am confused about path connectedness of $A$. Please help me about path connectedness of $A.$ Thanking you.
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