Continuity Of Functions
- 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