mathwonk said:try it on R. can you think of a function which gets larger somewhere inside the unit interval than it is at 1, -1? maybe you meant linear. or maybe functional means linear. then it seems true.
A sup of a continuous function is the smallest upper bound of the function's values. In other words, it is the largest value that the function can take on within a given interval.
To calculate the sup of a continuous function, you would need to find the maximum value of the function within the given interval. This can be done by finding the critical points of the function and evaluating the function at those points, as well as evaluating the function at the endpoints of the interval.
Yes, the sup of a continuous function can be infinite. This would occur when the function has no upper bound within the given interval, meaning that the function's values increase without limit.
The sup of a continuous function is the highest point on the function's graph within the given interval. It is also the point where the graph of the function begins to decrease, as the function's values cannot exceed the sup.
The sup of a continuous function is important because it helps us understand the behavior of the function within a given interval. It can also be used to determine if the function has a maximum value within the interval, as well as to evaluate limits and continuity of the function.