We consider a channel with a binary input X being corrupted by a continuous-valued noise that results in a continuous-valued output Y . An optimal binary quantizer is used to quantize the continuous-valued output Y to the final binary output Z to maximize the mutual information I(X; Z). We show that when the ratio of the channel conditional density r(y) = P (Y =y|X=0)/P (Y =y|X=1) is a strictly increasing or decreasing function of y, then a quantizer having a single threshold can maximize mutual information. Furthermore, we show that an optimal quantizer (possibly with multiple thresholds) is the one with the thresholding vector whose elements are all the solutions of r(y) = r* for some constant r* > 0. In addition, we also characterize necessary conditions using fixed point theorem for the optimality and uniqueness of a quantizer. Based on these conditions, we propose an efficient procedure for determining all locally optimal quantizers, and thus, a globally optimal quantizer can be found. Our results also confirm some previous results using alternative elementary proofs.