- #1
tellmesomething
- 411
- 54
- Homework Statement
- Let
for all where is continuous function & if then may be?
- Relevant Equations
- None
So I know that since that means can achieve all possible values on the real number line meaning is a constant function. And I know hwo to calculate the limit beyond that. However my teacher made a point which I dont necessarily agree with he said, if wasn't continuous we could not have said Its a constant function.
He gave an example like
for all
for all
The above piecewise function also satisfies
But my doubt is since the original question already says that , isnt that enough information to conclude its a constant function.
What I mean is if the question was instead
Let for all & , if then may be?
Could we still not have concluded that is a constant function I.e on a graph it would be parallel to the axis for alll ?
Because for all I know, a continuous function means that the limiting value and the functional value are equal , and here since we already know that the function for all we know that the limiting value and the functional value are equal..
He gave an example like
The above piecewise function also satisfies
But my doubt is since the original question already says that
What I mean is if the question was instead
Let
Could we still not have concluded that
Because for all I know, a continuous function means that the limiting value and the functional value are equal , and here since we already know that the function
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