The sine curve together with the y-axis is an example of a connected set that is not path-connected. It turns out that if a set is open as well as connected then it is if fact path-connected. Do not attempt to construct paths! Instead learn this amazing trick that takes local results to global results for free!
PDF here: https://drive.google.com/file/d/1WWW5xxZxeVcVa-Oi_9ENgc7y9MbMKIg0