The domain of definition of the function $$ y = \frac{1}{\log_{10}(1 - x)} + \sqrt{x + 2} $$

For the square root term, we need $$x+2 \ge 0$$ so $$x \ge -2$$ For the logarithm, its argument must be positive: $$1-x>0$$ so $$x<1$$ Also, the logarithm is in the denominator, so it cannot be zero: $$\log_{10}(1-x) \ne 0$$ This happens when $$1-x \ne 1$$ so $$x \ne 0$$ Combining all conditions, $$-2 \le x < 1,\ x\ne 0$$ Hence the domain is $$[-2,1)-\{0\}$$ This matches option $\mathrm{C}$.