the validity of (ε, δ)-definition of limit

Question

People use this definition by constructing δ for arbitrarily small ε, or proving that for some ε, δ does not exist.

So my question is: Is there a function that one can not construct or disprove the existence of δ for ε?

Answer 1

Take any statement $P$ that is logically undecidable, that is, it is not possible to either prove or disprove $P.$

Define a function $f$ as follows:

$$ f(x) = \begin{cases} 0 & \text{if $x$ is rational or $P$ is true} \\ 1 & \text{if $x$ is irrational and $P$ is false}. \end{cases} $$

Then $f$ is continuous at $0$ (in fact, $f(x)=0$ everywhere) if $P$ is true, but $f$ is not continuous at $0$ if $P$ is false. In one case it would be easy to construct $\delta$ and in the other it would be easy to show no such $\delta$ exists for $0 < \epsilon < 1.$ But it is impossible to prove which case holds, so we are unable to construct a suitable $\delta$ and unable to show it does not exist.

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Basically, your question is, "Are there problems that cannot be solved?" And the answer is, "Yes."