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What about the function $f:\omega_1\to\{0,1\}$ assigning to each ordinal $\alpha$ written in the Cantor normal form $\alpha=\omega^{\beta_1}c_1+\cdots\omega^{\beta_k}c_k$ the number $f(\alpha):=c_k\mod 2$?

It seems that the restriction of $f$ to a suitably large countable ordinal (which is a Polish scattered space) can have arbitrarily high oscillation rank $\beta(f)$.

In particular, $f|(\omega^\omega+1)$ has the oscillation rank $\beta(f)=\omega+1$.

Since the space $\omega_1$ is scattered, for every non-empty subset $A\subset \omega_1$ the restriction $f|A$ has a continuity point (which is a bit better than being of the first Baire class).

What about the function $f:\omega_1\to\{0,1\}$ assigning to each ordinal $\alpha$ written in the Cantor normal form $\alpha=\omega^{\beta_1}c_1+\cdots\omega^{\beta_k}c_k$ the number $f(\alpha):=c_k\mod 2$?

It seems that the restriction of $f$ to a suitably large countable ordinal (which is a locally compact Polish scattered space) can have arbitrarily high oscillation rank $\beta(f)$.

In particular, $f|(\omega^\omega+1)$ has the oscillation rank $\beta(f)=\omega+1$.

Since the space $\omega_1$ is scattered, for every non-empty subset $A\subset \omega_1$ the restriction $f|A$ has a continuity point (which is a bit better than being of the first Baire class).

What about the function $f:\omega_1\to\{0,1\}$ assigning to each ordinal $\alpha$ written in the Cantor normal form $\alpha=\omega^{\beta_1}c_1+\cdots\omega^{\beta_k}c_k$ the number $f(\alpha):=c_k\mod 2$?

It seems that the restrictions of $f$ to a suitably large countable ordinal (which is a Polish scattered space) can have arbitrarily high oscillation rank $\beta(f)$.

Since the space $\omega_1$ is scattered, for every non-empty subset $A\subset \omega_1$ the restriction $f|A$ has a continuity point (which is a bit better than being of the first Baire class).

What about the function $f:\omega_1\to\{0,1\}$ assigning to each ordinal $\alpha$ written in the Cantor normal form $\alpha=\omega^{\beta_1}c_1+\cdots\omega^{\beta_k}c_k$ the number $f(\alpha):=c_k\mod 2$?

It seems that the restriction of $f$ to a suitably large countable ordinal (which is a Polish scattered space) can have arbitrarily high oscillation rank $\beta(f)$.

In particular, $f|(\omega^\omega+1)$ has the oscillation rank $\beta(f)=\omega+1$.

Since the space $\omega_1$ is scattered, for every non-empty subset $A\subset \omega_1$ the restriction $f|A$ has a continuity point (which is a bit better than being of the first Baire class).

What about the function $f:\omega_1\to\{0,1\}$ assigning to each ordinal $\alpha$ written in the Cantor normal form $\alpha=\omega^{\beta_1}c_1+\cdots\omega^{\beta_k}c_k$ the number $(c_k\mod 2)$?

It seems that the restrictions of $f$ to a suitably large countable ordinal (which is a Polish scattered spaces) can have arbitrarily high oscillation rank $\beta(f)$.

Since the space $\omega_1$ is scattered, for every non-empty subset $A\subset \omega_1$ the restriction $f|A$ has a continuity point (which is a bit better than being of the first Baire class).

What about the function $f:\omega_1\to\{0,1\}$ assigning to each ordinal $\alpha$ written in the Cantor normal form $\alpha=\omega^{\beta_1}c_1+\cdots\omega^{\beta_k}c_k$ the number $f(\alpha):=c_k\mod 2$?

It seems that the restrictions of $f$ to a suitably large countable ordinal (which is a Polish scattered space) can have arbitrarily high oscillation rank $\beta(f)$.

Since the space $\omega_1$ is scattered, for every non-empty subset $A\subset \omega_1$ the restriction $f|A$ has a continuity point (which is a bit better than being of the first Baire class).