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bhsmith
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Homework Statement
Is t^2, -2t and 2 continuous on an open interval?
Homework Equations
I have re read the theorems and explanations of how something is continuous but i still don't understand it.
bhsmith said:Homework Statement
Is t^2, -2t and 2 continuous on an open interval?
Homework Equations
I have re read the theorems and explanations of how something is continuous but i still don't understand it.
The Attempt at a Solution
Continuous on an open interval means that the function is defined and has a value for every point within the interval, and there are no breaks or gaps in the graph of the function within that interval.
The main difference is that a function can be continuous on an open interval even if it is not continuous at the endpoints of the interval. In contrast, for a function to be continuous on a closed interval, it must also be continuous at the endpoints.
Continuous functions on open intervals have many useful properties and are essential in many mathematical and scientific applications. They allow us to make predictions and draw conclusions about the behavior of a function without having to evaluate every single point within the interval.
To determine if a function is continuous on an open interval, we can use the three-part definition of continuity. This means checking if the function is defined at every point within the interval, if the limit of the function exists at every point in the interval, and if the limit is equal to the value of the function at that point.
Yes, a function can be continuous on one open interval but not on another. This is because continuity depends on the behavior of the function within the interval, and different intervals may have different behaviors. For example, a function may be continuous on the interval (0,1) but not on the interval (1,2).