Задание №1.

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