Determine if the equation defines y as a function of x: 2xy=1

Solve for y...

2xy=1

y=1/(2x)

So this is a function as each x value (with the exception of x=0) produces just one value for y.  However, technically there is a vertical asymptote about the vertical line x=0.  So there is a discontinuity as x=0.

I say this technically, because most would say that this function is true even with the discontinuity (as they do with say y=tana), but technically the discontinuity makes this not a function because another requirement of a function is for their to be an actual output value for each input value.  In this case division by zero when x=0 is undefined and cannot be included as part of a function.  So if you are picky about being absolute, 2xy=1 is really two functions.  One with a domain of (-oo,0) and another with a domain of (0,+oo).