The function f(x) is defined as f(x) = x - 1 for x < 0 and f(x) = x + 1 for x ≥ 0. It is proven that f: R -> R is strictly increasing by applying the definition of a strictly increasing function, which states that for any x1 < x2, f(x1) < f(x2). Additionally, it is established that the inverse function f^(-1): f(R) -> R is continuous at the point 1, using the definition of continuity.
PREREQUISITESStudents studying calculus, mathematicians interested in function properties, and educators teaching concepts of increasing functions and continuity.
Hello nikefish. Welcome to PF !nikefish said:Homework Statement
Hello, Does anybody know how to solve this question? Or a formal way to prove that a fuction is strictly increasing?
Define f(x)={ x-1 if x<0
x+1 if x >_ 0
Show that f: R -> R is strictly increasing and that f^(-1) : f(R) -> R is continuous at 1
Homework Equations
The Attempt at a Solution
nikefish said:Homework Statement
Hello, Does anybody know how to solve this question? Or a formal way to prove that a fuction is strictly increasing?
Define f(x)={ x-1 if x<0
x+1 if x >_ 0
Show that f: R -> R is strictly increasing and that f^(-1) : f(R) -> R is continuous at 1
Homework Equations
The Attempt at a Solution