vikkisut88 said:Homework Statement
Suppose that f: R -> R is continuous on R and that lim (x -> [tex]\infty[/tex]+)(f(x) = 0) and lim (x -> [tex]\infty[/tex]-)(f(x)=0).
Prove that f is bounded on R
vikkisut88 said:Homework Equations
I have got the proof of when f is continuous on [a,b] then f is bounded on[a,b] but I'm unsure as to whether I can use this proof for this particular question.
A function is considered bounded on the real numbers (denoted by R) if there exists a specific number M, such that the absolute value of the function's output is always less than or equal to M for all input values on the real number line. In other words, the function never reaches values that are infinitely large or small.
To prove that a function is bounded on R, you must show that there exists a specific value M that satisfies the definition of boundedness. This can be done by using various mathematical techniques, such as finding the maximum or minimum values of the function, or using the squeeze theorem.
If a function is not bounded on R, it means that there is no limit to how large or small the function's output can be. This can lead to problems in certain mathematical calculations and can make it difficult to make predictions based on the function's behavior.
Yes, it is possible for a function to be bounded overall on R, but not bounded on a specific interval within R. This means that while the function may have a limit on R, it may not have a limit on a smaller subset of R.
Yes, there are different types of boundedness for functions on R. A function can be bounded above, meaning there is a maximum value for its output, or bounded below, meaning there is a minimum value for its output. Additionally, a function can be both bounded above and below, meaning there is both a maximum and minimum value for its output.