Arctan(x) is the inverse function of tangent, denoted as tan^-1(x). It is also known as the arc tangent function and is used to find the angle whose tangent is x.
To show that arctan(x) exists, we need to prove that the inverse function of tangent exists for all real numbers. This can be done by showing that the tangent function is one-to-one and onto, meaning it has a unique output for every input and that every real number has a corresponding input in the tangent function.
The domain of arctan(x) is all real numbers, while the range is limited between -π/2 and π/2, or -90 degrees and 90 degrees, respectively.
The continuity of arctan(x) can be demonstrated by showing that it is differentiable for all real numbers. This can be done by proving that the tangent function, which is the derivative of arctan(x), is continuous for all real numbers.
It is important to show that arctan(x) exists and is continuous because it allows us to use this function in various mathematical calculations and applications. It is also a fundamental concept in trigonometry and calculus.