- #1
Ric-Veda said:Assume that f is a continuous, real-valued function defined on a metric space X. If {xn} is a sequence in X converging to x, prove that Limn→∞f(xn) = f(x).
Here is my attempt, but I am not sure if it is correct.
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Ricster55 said:Here is my new solution, but I'm not sure if the graph I put is of any use, but if you can, please put up a similar graph to the one I put to help better understand.
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A continuous function is one where the output changes smoothly as the input changes. This means that there are no sudden jumps or breaks in the graph of the function. In other words, if you were to draw the graph of the function without lifting your pencil, it would be a continuous line.
A function f is continuous at a point a if the limit of f(x) as x approaches a exists and is equal to f(a). This means that the value of the function at a is equal to the value it approaches as x gets closer and closer to a.
Yes, it is possible for a function to be continuous at some points and not others. This is known as a piecewise continuous function. It means that the function may have different behaviors or rules at different intervals, but each of these intervals are continuous.
Continuity and differentiability are two different concepts. A function is continuous if it has no breaks or jumps, while a function is differentiable if it has a well-defined derivative at every point. In other words, a differentiable function is always continuous, but a continuous function may not necessarily be differentiable.
The intermediate value theorem states that if a continuous function f(x) takes on two different values at points a and b, then it must also take on every value in between at some point c between a and b. This theorem is only applicable to continuous functions, as it relies on the concept of a continuous graph without any breaks or jumps.