Answer
The function is discontinuous at $x = 0$.
Work Step by Step
$\lim\limits_{x \to 0^{-}}\frac{1}{1+e^{\frac{1}{x}}} = \frac{1}{1+e^{-\infty}} = \frac{1}{1+0} = 1$
$\lim\limits_{x \to 0^{+}}\frac{1}{1+e^{\frac{1}{x}}} = \frac{1}{1+e^{\infty}} = \frac{1}{1+\infty} = 0$
Since both limits are not the same, we can conclude that $y = \frac{1}{1+e^{\frac{1}{x}}}$ is not continuous at $x = 0$.
