The function f(x) = [x], where [x] is the greatest integer function that is less than or equal to x, is continuous at
An integer is a whole number that can be positive, negative, or zero, without fractions or decimals. It includes numbers like -3, 0, 7, and 42. Integers are used in counting, ordering, and algebra. They follow arithmetic operations and are essential in mathematics, computing, and real-world applications.
Class 12 Maths Chapter 5 Continuity and Differentiability is an important topic for the CBSE Exam 2024-25. It covers the concepts of continuity of functions differentiability rules differentiation of inverse trigonometric functions and second-order derivatives. Understanding these concepts helps in solving calculus problems and builds a strong foundation for advanced mathematics.
Share
The function f(x) = ⌊x⌋, where ⌊x⌋ is the greatest integer function (also known as the floor function), gives the greatest integer less than or equal to x.
Continuity of the floor function:
The floor function is discontinuous at integer values of x. This is because, at any integer n, the function jumps from n-1 to n. Hence, the function is not continuous at integer points.
Continuity at non-integer points:
At non-integer points, the function is continuous since it is a constant between integers.
Checking the given points:
– At x = 4, f(x) is not continuous since it jumps at an integer value.
– At x = -2, f(x) is also not continuous since it jumps at an integer value.
– At x = 1.5, f(x) = 1, which is continuous since it is not at an integer point.
– At x = 1, the function f(x) is discontinuous because it hops at the integer value.
Click here for more:
https://www.tiwariacademy.com/ncert-solutions/class-12/maths/#chapter-5