- #1

tellmesomething

- 393

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- Homework Statement
- Let ##f(x)=f(2x)## for all ##x\in R##where ##f(x)## is continuous function & ##f(2024)=π/2## if ##L=lim_{x\to 0} \frac{(cos²(f(x))+1-sin²(f(x)))} {sin²x}## then ##4L## may be?

- Relevant Equations
- None

So I know that since ##x \in R## that means ##2x## can achieve all possible values on the real number line meaning ##f(x)## 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 ##f(x)## wasn't continuous we could not have said Its a constant function.

He gave an example like

##f(x)=1## for all ##x \in Q##

##f(x)=0## for all ##x \not\in Q##

The above piecewise function also satisfies ##f(x)=f(2x)##

But my doubt is since the original question already says that ##x \in R##, isnt that enough information to conclude its a constant function.

What I mean is if the question was instead

Let ##f(x)=f(2x)## for all ##x \in R## & ##f(2024)=π/2##, if ##L=lim_{x\to 0} \frac{(cos²(f(x))+1-sin²(f(x)))} {sin²x}## then ##4L## may be?

Could we still not have concluded that ##f(x)## is a constant function I.e on a graph it would be parallel to the ##x## axis for alll ##x## ?

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 ##f(x)=f(2x)## for all ##x\in R## we know that the limiting value and the functional value are equal..

He gave an example like

##f(x)=1## for all ##x \in Q##

##f(x)=0## for all ##x \not\in Q##

The above piecewise function also satisfies ##f(x)=f(2x)##

But my doubt is since the original question already says that ##x \in R##, isnt that enough information to conclude its a constant function.

What I mean is if the question was instead

Let ##f(x)=f(2x)## for all ##x \in R## & ##f(2024)=π/2##, if ##L=lim_{x\to 0} \frac{(cos²(f(x))+1-sin²(f(x)))} {sin²x}## then ##4L## may be?

Could we still not have concluded that ##f(x)## is a constant function I.e on a graph it would be parallel to the ##x## axis for alll ##x## ?

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 ##f(x)=f(2x)## for all ##x\in R## we know that the limiting value and the functional value are equal..

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