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
A series is considered strictly increasing if each term in the series is larger than the previous term. In other words, the values in the series are always increasing and there are no repeats.
A function is considered strictly increasing if for any two values in the domain, the output value of the second value is greater than the output value of the first value. This can be determined by graphing the function or by finding the derivative and checking if it is always positive.
Strictly increasing functions are useful in many mathematical and scientific applications. They allow for easier analysis and prediction of data, and can also be used to model real-world situations such as population growth or financial investments.
No, a series or function cannot be both strictly increasing and decreasing. This is because the definitions of strictly increasing and decreasing are mutually exclusive - if a series is always increasing, it cannot also be always decreasing.
Yes, there are many real-life examples of strictly increasing functions. Some examples include the growth of a population, the increase in height of a growing tree, and the accumulation of interest on a savings account.