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Of course you want to rule out the constant function, so you probably mean that there is a unique highest and lowest point in the neighborhood. Assuming this, with your new definition of turning point, you can choose your neighborhoods to be intervals with rational endpoints. This will force the number of turning points to be countable.

BTW: Do you know about math.stackexchange.com? That is a better site for questions that are interesting but not quite research-level!

Of course you want to rule out the constant function, so you probably mean that there is a unique highest and lowest point in the neighborhood. Assuming this, with your new definition of turning point, you can choose your neighborhoods to be intervals with rational endpoints. This will force the number of turning points to be countable.

Of course you want to rule out the constant function, so you probably mean that there is a unique highest or lowest point in the neighborhood. Assuming this, with your new definition of turning point, you can choose your neighborhoods to be intervals with rational endpoints. This will force the number of turning points to be countable.

BTW: Do you know about math.stackexchange.com? That is a better site for questions that are interesting but not quite research-level!

Of course you want to rule out the constant function, so you probably mean that there is a unique highest and lowest point in the neighborhood. Assuming this, with your new definition of turning point, you can choose your neighborhoods to be intervals with rational endpoints. This will force the number of turning points to be countable.

BTW: Do you know about math.stackexchange.com? That is a better site for questions that are interesting but not quite research-level!

Of course you want to rule out the constant function, so you probably mean that there is a unique highest or lowest point in the neighborhood. Assuming this, with your new definition of turning point, I'm pretty sure you can choose your neighborhoods to be intervals with rational endpoints. This will force the number of turning points to be countable.

BTW: Do you know about math.stackexchange.com? That is a better site for questions that are interesting but not quite research-level!

Of course you want to rule out the constant function, so you probably mean that there is a unique highest or lowest point in the neighborhood. Assuming this, with your new definition of turning point, you can choose your neighborhoods to be intervals with rational endpoints. This will force the number of turning points to be countable.

BTW: Do you know about math.stackexchange.com? That is a better site for questions that are interesting but not quite research-level!