Is it permissible to use the internet on a mobile phone without the knowledge and consent of one's father, knowing that the usage is only for what Allah has made permissible? And if it is not permissible, is it obligatory to inform him of past usage?

Obedience to parents is obligatory after obedience to Allah, as long as they do not command disobedience. If a parent forbids the use of the internet and there is a benefit for him in that, or if he is the one paying for its use, then it is not permissible to disobey him, and it is obligatory to inform him and seek his permission.

However, if there is no benefit for him in the prohibition and no harm results to him from the use, then it is permissible to use it without his knowledge if the use is for good and brings about benefits.

The criterion for the obligation of obedience to parents is that they command something that benefits them. But if they command something in which there is no benefit for them, or which causes harm, then their obedience is not obligatory. However, one must deal with them with kindness and good conduct. The child is not considered disobedient if there is a benefit in disobeying them. If the benefit clearly lies in obeying them, then there is goodness and blessing in it.