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I perfectly know that a full circle is not a function. I just want to know that is there any condition under which a circle becomes a function and if there is, then what is it ?

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    $\begingroup$ There are many wording issues in this question. A circle, or half a circle, strictly speaking is not a function, it is just a subset of the plane. The graph of $f(x)=\sqrt{1-x^2}$ over $[-1,1]$ is half a circle, true, but the point is that a function is a function if it is a function, i.e. if it maps every $x$ in some domain into a unique $f(x)$. $\endgroup$ Commented Jan 12, 2017 at 16:28

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Only circles with infinite or zero curvature are explicit on the reals. Otherwise the vertical line test would fail.

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In polar coordinates, a circle is the graph of the function $r=a$.

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