The document discusses continuity of functions. It provides three conditions for a function f(x,y) to be continuous at a point (a,b): 1) f(x,y) must be defined at (a,b), 2) the limit of f(x,y) as (x,y) approaches (a,b) must exist, and 3) this limit must equal f(a,b). It then shows that the function f(x,y)=x^2+2y is continuous at the point (1,2), but the function f(x,y)=2xy/(x^2+y^2) is discontinuous at the origin (0,0)

1. Gandhinagar Institute Of
Technology(012)
Active Learning Assignment
Subject:- Calculus (2110014)
Topic:- Continuity OF Function
Branch:- C. E.
Batch:-
Prepared by:-
Enrollment no:
Guided by:-

2. Introduction
• A function is said to be continuous at x=a if
there is no interruption in the graph of f(x) at a.
And its graph is unbroken at a, and there is no
hole, jump or gap in the graph.

3. Continuity
• A function is called continuous at c if the
following three conditions are met:
1. f(a,b) exists, i.e.,f(x,y) is defined at (a,b).
2. exists.
3. .
( , ) ( , )
lim ( , )
x y a b
f x y

( , ( , )
lim ( , ) ( , )
x y a b
f x y f a b



4. Example
Show that f(x,y)= 𝑥2
+ 2y is continuous at (1,2).
Solution
lim
𝑥,𝑦 →(1,2)
𝑓(𝑥, 𝑦)= lim
𝑥,𝑦 →(1,2)
(𝑥2
+ 2y ) = 12
+ 2(2) = 5
𝑓(1,2)=12
+ 2(2) = 5
lim
𝑥,𝑦 →(1,2)
𝑓(𝑥, 𝑦)= 𝑓(1,2)
Hence, f(x,y) is continuous at (1,2).

5. Example
Show that f(x,y) =
2𝑥𝑦
𝑥2+𝑦2 (x,y) ≠ (0,0)
= 0 (x,y) = (0,0)
is continuous at every point except at the origin.
Solution
lim
𝑥→0
lim
𝑦→0
𝑓 𝑥, 𝑦 = lim
𝑥→0
lim
𝑦→0
2𝑥𝑦
𝑥2+𝑦2
= lim
𝑥→0
0 = 0
lim
𝑥→0
lim
𝑦→0
𝑓 𝑥, 𝑦 = lim
𝑥→0
lim
𝑦→0
2𝑥𝑦
𝑥2+𝑦2
= lim
𝑥→0
0 = 0
Putting y = mx and taking limit x →0,
lim
𝑥→0
2𝑥(𝑚𝑥)
𝑥2+(𝑚𝑥)2
= lim
𝑦→0
2𝑚
1+𝑚2 =
2𝑚
1+𝑚2

6. Since the last limit depends on m and m is not fixed, the limit does not
exist.
Hence, f(x) is discontinuous at origin i.e., (0,0).
Let (x,y) = (a,b) ≠ (0,0) be an arbitrary point in xy-plane, where a
and b are real numbers.
lim
𝑥,𝑦 →(𝑎,𝑏)
𝑓(𝑥, 𝑦)= lim
𝑥,𝑦 →(𝑎,𝑏)
2𝑥𝑦
𝑥2+𝑦2
=
2𝑎𝑏
𝑎2+𝑏2
f(a,b)=
2𝑎𝑏
𝑎2+𝑏2
which is finite for real values of a and b.
lim
𝑥,𝑦 →(𝑎,𝑏)
𝑓 𝑥, 𝑦 = 𝑓(𝑥, 𝑦)
This shows that f(x,y) is continuous at (a,b).
Hence, f(x,y) is continuous at every point except the origin.

7. If f and g are functions, continuous at x = c
Then …
f(x) + g(x) is continuous
is continuous
is continuous
f(g(x)) is continuous
( ) ( )f x g x
( )
( )
f x
g x