Hindsight being 2020, might it have been better to make the block reward function a more continuous function rather than quadrennial steps?

The fact that the halving schedule exists is important, but the importance is that it is a converging function. If the halving was not a change of block reward, but a change to the rate of change to the block reward, the 4-year cycle would be calmed. Maybe the supply shock is actually good for price action, and that price action is good for adoption. The world will never know.

Just for fun, I came up with a few alternate block subsidy schedules.

Imagine that the very first block reward starts at 100 units, and the subsidy decreased with each block by 90μ units until block height 1M, at which time the subsidy is 10 units. Then, the subsidy decreases by 9μ each block for the next 1M-block epoch, and this pattern repeats we run out of bits (the unit is biased by 10 or so digits, of course).

Subsidy ~= 10^(2 - epoch) - 10^(1 - epoch) * (blocks in epoch)/(epoch length)

This is very "base-10"-ish and not very Bitcoin-y, so we could take an inverse approach and calculate a block height that would result in a halving schedule approximating Bitcoin's, just as a continuous function.

Br: Block Reward

Bb: Block Bias

Bh: Block Height

El: Epoch Length

Eb: Epoch Bias

E: Epoch

S: Scale

Br = (Bb-Bh) % El * 2^(Eb - E) + El*S*(2^(Eb - E) - 1)

Eb = floor( Bb / El )

E = Eb - floor( (Bb-Bh) / El )

If you set the Block Bias to 3140165, and the Scale to 10^-8, the Epoch Length being the 210000, same as Bitcoin's, the supply cap is 20999993.36, pretty darn close to Bitcoin's.

I'm just a little disappointed that the block height bias didn't turn out to be π-million. That would have been an amazing! Pi turns up just so mathemagically often.