Limits Of Exotic Functions

Calculus Level 3

f(x)={0 if x is irrational1 if x is rationalg(x)={0 if x is irrational1q if x=pq, where p and q are coprime nonnegative integers \begin{aligned} f(x) &= \begin{cases} 0 &\text{ if } x \text{ is irrational} \\ 1 &\text{ if } x \text{ is rational} \end{cases} \\ g(x) &= \begin{cases} 0 &\text{ if } x \text{ is irrational} \\ \frac1q &\text{ if } x =\frac{p}{q}, \text{ where } p \text{ and } q \text{ are coprime nonnegative integers} \end{cases} \end{aligned}

Let f(x) f(x) and g(x) g(x) be two functions defined on [0,1] [0,1] by the formulas as described above.

For which a(0,1) a \in (0,1) does the (deleted) limxaf(x) \lim\limits_{x\to a} f(x) exist?

For which b(0,1) b \in (0,1) does the (deleted) limxbg(x) \lim\limits_{x\to b} g(x) exist?


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