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Ted123
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What is an example of a continuous function [itex]f:\mathbb{R}\to\mathbb{R}[/itex] such that [itex]f(\mathbb{R})[/itex] is open?
CompuChip said:Well, [itex]\mathbb{R}[/itex] is open in [itex]\mathbb{R}[/itex], right?
A continuous function with open range is a function where the range of values it can take on is an open interval, meaning it does not include its endpoints. This means that the function can take on any value within the interval without any gaps or jumps.
A continuous function with open range allows for the function to take on any value within the interval, while a continuous function with closed range only allows for the function to take on values within the interval including its endpoints. This means that a continuous function with open range has a wider range of possible values.
Continuity in a function with open range ensures that there are no gaps or jumps in the values that the function can take on within the interval. This allows for a smooth, uninterrupted graph and makes it easier to analyze the behavior of the function.
The continuity of a function with open range is determined by checking if the limit of the function at each point within the interval is equal to the value of the function at that point. If the limit and function value are equal, the function is continuous at that point.
Continuous functions with open range are commonly used in modeling natural phenomena such as population growth, temperature changes, and fluid dynamics. They are also used in economics and finance to model stock prices and interest rates.