Continuous image of a locally connected space which is not locally connected The question is pretty much in the title, I'm looking for an example of a locally connected space and continuous mapping such that the image is not locally connected. Thanks! EDIT: Corrected the phrasing to the intended meaning.

Consider the following variant on the topologist's sine curve.

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This space $X$ consists of the graph of $y = \sin(\pi/x)$ for $0
However, there exists a continuous surjection $f\colon [0,2)\to X$. Specifically, $f(0) = (0,0)$ and $f(1) = (1,0)$, with $f(t)$ following along the bottom curve for $0\leq t\leq 1$. For $t>1$, the function follows along the sine curve, i.e. $$ f(t) \;=\; \left(2-t,\sin\left(\frac{\pi}{2-t}\right)\right)\qquad\text{for }t> 1\. $$