I
(2) (a) Let f : [0,1] → R be a continuous function such that f(1) = f(VI) for all
re (0,1). Show that f is constant.
[4]
(b) Let f (r)
o if req
rf €RNO
Determine the set of points in which f is
continuous.
(5)
Show that there exists
(c) Let : (0,0) = R be defined by f(0) = 3
ro € (0.00) such that f(ro) =
(3)
Answers
Step-by-step explanation:
and today I’m here with a holiday story to commemorate Hanukkah, the Jewish Festival of Lights.
Hanukkah celebrates the rededication of the Temple of Jerusalem over 2000 years ago.
First a tiny bit of history.
After the time of Alexander the Great, much of the middle East was ruled by Greek Kings. One of these, Antiochus IV, forbade the Jews from carrying out their religion. A group of Jewish warriors called the Maccabees led a revolt against Antiochus in the year 167 BCE . When they retook Jerusalem, they rededicated the Temple. They wanted to relight the oil lamps. Although they could only find a tiny bit of oil, the oil miraculously lasted for a full 8 days. This was a miracle.
The symbol of Hanukkah is a candelabra called the Menorah. Each night of the festival, one candle is lit as the blessing is spoken. There are 8 days in all, but there are 9 candles. The ninth, which stands in the centre, is used to light the others.
Hanukkah is a time of family fun and celebration. A traditional game is to spin a dreidel, a kind of spinning top with Hebrew letters on it. Prizes are often chocolate “gelt” or money. Hanukkah is also associated with festive food such as donuts and creamy fried potato and onion balls called Latkes.
Food features in the ancient Jewish story which I have chosen. The main character is the wife of a rabbi or priest. It’s called, The Nosy Neighbour.
MOST of the rabbis were very poor. But the poorest of all was Haninah ben Dosah. Even on Friday, his wife had nothing to cook and nothing to bake. You know how busy your mother is on Friday. You know what good odors come out of the kitchen. You can smell the freshly baked pies, donuts, and cookies. You can smell the chicken soup. But the wife of Ben Dosah didn't even have to build the fire, for she had nothing to cook on it.
"Dear me!" she said. "All the women around me are busy getting ready for the Sabbath. Only my home is cold and empty. I don't want them to pity us. Who can tell? Some kind neighbour may even bring us some food. O dear! O dear! What shall I do?"
You know at that time they did not have gas ranges or even coal stoves. They had ovens made of earthen clay. So Ben Dosah's wife took some wood and built a fire. The wood burned and the smoke went up through the chimney.
It didn't take long before all the neighbours were talking about the wonderful thing that had happened. Haninah's wife had something to cook and to bake. And most of them were happy.
Now Ben Dosah's wife had one extremely nosy neighbour. When she saw the smoke coming out of the chimney, she thought:
"The Rabbi's wife has nothing in her oven. What is all that smoke about? I must go in to see what she is burning in her oven." So the nosy neighbor threw her shawl over her shoulders and walked across to the Rabbi's house. She knocked at the door and waited a while. The Rabbi's wife, however, was ashamed to meet her. Therefore, instead of answering the knock, she ran into the next room. Again the nosy neighbour knocked at the door.
"O dear!" she thought. "I guess she is ashamed to meet me. Well, I'll go in anyway." So without waiting for any answer to her knock, she quickly opened the door and went into the house.
What do you think she saw? There on the stove all kinds of latkes and festive dishes were being prepared. And she could smell cherry pies and honey cookies in the stove. It even seemed to her that she smelled the cookies burning.
When she saw all this she called to the Rabbi's wife, "Hurry up. Your cookies are burning."
"What was that? What did you say?" the Rabbi's wife called from the other room. She was sure that she hadn't heard right, because she knew that there was nothing in the stove. "Is it possible that she is making fun of me?" the Rabbi's wife thought.
"Well, what are you waiting for? Hurry before they are all burnt," the neighbour's voice was heard again.
The Rabbi's wife, full of joy, ran into the kitchen. She was blushing. But it didn't matter. She looked as if she were hot because of the hard work. Going over to the stove, she touched the cookies to make sure that they were real, and said:
"'I had just run out to get my wooden spoon when you knocked on the door."