Jesus: Sabbath in Capernaum
After the events in Nazareth, Jesus went on to Capernaum, and again goes into the synagogue on the Sabbath to teach. It is noted by the hearers that his message has authority. However, it is interupted by a demon-possessed man who shouts loudly. Jesus speaks to the demon and it leaves the man. There are no apparent protests; everyone is amazed by the authority of his teaching. This incident is also related in Mark 1:21-28, but in that account the people question if his teaching is "new". Jesus answers that question decisively not long after.
In both Mark and Luke Jesus leaves the synagague and goes to the home of Simon, where he heals Simon's mother-in-law. (Note that Simon was married and had a residence in Capernaum.) It is still the Sabbath; there are no complaints about his healing.
In Jewish tradition, based on biblical teaching, the Sabbath lasts from sundown on Friday until sundown on Saturday. It is interesting that as the passage continues, people wait until after sunset before bringing him sick people to heal. The issue wasn't so much the healing on the Sabbath as it was the carrying of people. Heavy loads were not to be carried on the Sabbath. Here is an instance where the healing can be done just as well on another day, and Jesus does so.
In the following chapters he does a lot of teaching and healing that is not on the Sabbath. We know this because we have fishermen washing their nets and fishing, and men carrying a sick friend. These things would not have been done on the Sabbath, by Jesus or anyone else in that society.