Can logistic regression take continuous probability as label? Let's consider logistic regression for binary classification, with label 0 or 1. The loss function is -ylog(x) - (1-y)log(1-x), where x is predicted probability for label 1 and y is the label. In sklearn, logistic regression only take discrete labels. Why can't y be a continuous value between [0, 1]? Theoretically, Is there any mathematical problem if I label my samples like 0.75 being label 1 and 0.25 chance being label 0?
The question is mostly inspired by the implementation of logistic regression in sklearn, which does not take continuous input. For continuous input, every distinct number is considered as a class https://github.com/scikit-learn/scikit-learn/blob/0d378913b/sklearn/linear_model/_logistic.py#L1517.

of course it can. you have your inputs $x_{1,k}$, $x_{2,k}$, ..., $x_{n,k}$ and your output is: $y_k=1/(1+\exp(\beta_0+\beta_1x_{1,k}+\beta_2x_{2,k}+...+\beta_nx_{n,k}))$,
where $k$ are different measurements and $\beta_i$ are unknown parameters. You can now train with any $y_k \in [0,1]$, not only 0/1 labels (0,1 values). Newton iterations + partial derivatives will find you your optimal values of parameters $\beta_i$. Your optimization function can be for example $J=\sum_k{\large(y_k-1/(1+\exp(\beta_0+\beta_1x_{1,k}+\beta_2x_{2,k}+...+\beta_nx_{n,k}))\large)^2}$
