Formally, for an input $x$, $D(x)$ gives you the probability of $x$ being real. In this sense $D:\mathcal{X}\rightarrow [0,1]$, where $\mathcal{X}$ is the input space.
That said, the output of the discriminator is a probability (hence within 0 and 1), and you get the prediction (fake or real) by considering the most probable outcome. Informally $D(x) = fake$ for $D(x) < 0.5$ and real otherwise.
@edit: on the issue of Least Squares GAN.
In the first 2 paragraphs of my answer, I considered the case of the original GAN proposed by Goodfellow et al. [1]. Nonetheless other types of GANs exist, that do not employ a sigmoid activation at the output layer of the discriminator. That is the case of the Least Squares GAN of Mao et al. [2], upon which the authors of CycleGAN based themselves on [3].
The authors of [2] raise the issue of using a sigmoid activation in section 3.2:
when updating the generator, this loss function (cross-entropy on sigmoid activations) will cause the problem of vanishing gradients for the samples that are on the correct side of the decision boundary, but are still far from the real data.
The LS-GAN proposes the following workaround: $D(x) \in \mathbb{R}$, thus no sigmoid activation at the end. The loss is then substituted by the least squares loss. As follows, there is no clear encoding for fake and real labels anymore. For that reason the authors of [2] introduce constants $a$ and $b$, such that if $D(x) \approx a \implies x$ is fake, and $D(x) \approx b \implies x$ is real. For the generator, there is another constant $c$ such that $D(G(z)) \approx c \implies G(z)$ is fake. In [3] the authors picked $a = 0$, $b = 1$ and $c = 1$.
References
[1] Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., ... & Bengio, Y. (2014). Generative adversarial nets. Advances in neural information processing systems, 27.
[2] Mao, X., Li, Q., Xie, H., Lau, R. Y., Wang, Z., & Paul Smolley, S. (2017). Least squares generative adversarial networks. In Proceedings of the IEEE international conference on computer vision (pp. 2794-2802).
[3] Zhu, J. Y., Park, T., Isola, P., & Efros, A. A. (2017). Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision (pp. 2223-2232).