2 êóðñ (íåïðåðûâíîñòü è äèôôåðåíöèðóåìîñòü) Èññëåäîâàòü íåïðåðûâíîñòü ôóíêöèè ïðè x=0, y=0: 1. f (x, y) = x2 y 3 ; f (0, 0) x2 +y 2 =0 2. f (x, y) = x3 y 2 ; f (0, 0) x2 +y 2 =0 3. f (x, y) = 2x2 y 2 ; f (0, 0) x2 +y 2 =0 4. f (x, y) = 2x3 y 3 ; f (0, 0) x2 +y 2 =0 5. f (x, y) = x4 −y 4 ; f (0, 0) x4 +y 4 =0 6. f (x, y) = xy ; f (0, 0) x2 +y 2 =0 7. f (x, y) = 1 ; f (0, 0) x2 +y 2 =0 8. f (x, y) = x2 y 2 ; f (0, 0) x4 +y 4 =0 9. f (x, y) = 2xy ; f (0, 0) x2 +y 2 =0 10. f (x, y) = 2 ; f (0, 0) x2 +y 2 =0 11. f (x, y) = 3x2 y 2 ; f (0, 0) x4 +y 4 =0 12. f (x, y) = x2 −y 2 ; f (0, 0) x4 +y 4 =0 13. f (x, y) = x4 y 2 ; f (0, 0) x2 +y 2 =0 Äîêàçàòü, ÷òî ñëåäóþùèå ôóíêöèè íåïðåðûâíû â íà÷àëå êîîðäèíàò: ( 14. 15. 16. f (x, y) = √sin xy , x2 + y 2 6= 0 2 2 x +y 0, x2 + y 2 = 0. ( x3 ln(x2 + y 2 ), x2 + y 2 = 6 0 f (x, y) = 2 2 0, x + y = 0. ( 3 3 x +y x2 + y 2 6= 0 2 2, f (x, y) = x +y 0, x2 + y 2 = 0. 1 ( 17. 18. 19. f (x, y) = f (x, y) = f (x, y) = sin xy , x y, ( x 6= 0 x = 0. arctg x−arctg y , x−y 0, ( 3 x y−x2 y 2 , x3 −y 3 0, x 6= y x = y. x 6= y x = y. Äîêàçàòü, ÷òî ñëåäóþùèå ôóíêöèè íå ÿâëÿþòñÿ íåïðåðûâíûìè â íà÷àëå êîîðäèíàò: 20. 21. ( sin xy , f (x, y) = 0, ( 3 2 x +y 3 3, f (x, y) = x +y 0, ( 22. √x 2 +y x2 +y 2 f (x, y) = 0, ( 3 23. 24. 25. f (x, y) = f (x, y) = f (x, y) = x +y 2 , x2 +y 0, ( 2xy , x2 +y 2 0, ( 3 x +y 2 , x2 +y 2 0, x2 + y 2 = 6 0 2 2 x + y = 0. x2 + y 2 = 6 0 x2 + y 2 = 0. , x2 + y 2 6= 0 x2 + y 2 = 0. x2 + y 6= 0 x2 + y 2 = 0. x2 + y 2 = 6 0 x2 + y 2 = 0. x2 + y 2 = 6 0 x2 + y 2 = 0. 2