- #1
fibfreak
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Homework Statement
Let f:[a, b] -> R have a limit at each x in [a, b]. Prove that f is bounded.
Homework Equations
None
The Attempt at a Solution
No idea on how to start the proof. Completely lost.
Thank you
A function is said to be bounded if there exists a number M such that the absolute value of the function is always less than or equal to M, for all values of the independent variable.
To prove that a function is bounded, you need to show that the absolute value of the function is always less than or equal to a specific number for all values of the independent variable. This can be done using various techniques such as using the definition of a bounded function or using the properties of limits.
Proving that a function is bounded is important because it helps us understand the behavior of the function and its range of values. It also allows us to make predictions about the function and its limits, which can be useful in many applications of mathematics and science.
Some common strategies for proving that a function is bounded include using the squeeze theorem, applying the definition of a bounded function, and using the properties of limits such as the limit comparison test or the ratio test.
No, a function cannot be both bounded and unbounded. A function is either bounded or unbounded, depending on whether there exists a specific number that bounds the absolute value of the function for all values of the independent variable. A function cannot have both properties at the same time.