As a result, Jesus is telling the lady that, as a "Canaanite," she is like a dog since she will "eat" any sort of meat presented to her and assign equal value to it all. As a result, she is referred to as "a lady of enormous faith," although "faith" does not mean "merely to believe in." It means to have confidence in, to rely on.
This story takes place in Galilee, which was known for its fishing villages at the time. Since there were no cars or trucks, everything had to be carried by hand or on the back of a donkey. Fish would be cleaned immediately after being caught and would often be used before it even had a chance to smell bad.
The woman comes from Canaan, which was known for its vineyards at the time. Wine was one of the most important exports from Israel because without it, they could not have enjoyed the good life in Egypt. The woman sees Jesus as a great healer and brings him all her sick children so he can heal them. When Jesus says that only God can grant eternal life, she believes him to be the Son of David and asks him to come to her house.
In order for someone to be saved they need to believe in Jesus with their heart. Sometimes we give up too easily on people - we think they are lost when in fact they may just need to be shown love and mercy.
The Canaanite woman believed in God wholeheartedly. She was sure that the only way to obtain genuine mercy and total healing was via Jesus. In the gospel reading, Jesus stated that salvation was for the Jewish people. The woman's confidence astounded Jesus; it was amazing faith from a non-Israelite. However, what Jesus did not say is that only the Jews can be saved. His reply to her question showed that anyone who believes in him will be saved.
Faith is taking hope to its logical conclusion. It is believing that though God may not give you what you want when you want it, he will give you what you need when you need it. To the Canaanite woman, this must have seemed like madness since she had been healed but was still suffering from her illness. However, she knew that God always keeps his promises, so she took hope that she would be healed forever. This is why faith works with joy even when we don't feel like it can work with pain or disappointment. Our heavenly Father knows our heart, and he always comes through for those who trust him.
Jesus kept his promise to her and granted her request for a miracle. Her faith allowed her to see something that reality could have obscured from her view -- that everyone is capable of being saved. As long as you believe in Jesus' death on the cross and his resurrection, you will be saved.