What is a continuous mapping?

The challenge page says that “You may choose any mapping between the state vector domain D to the distribution domain (i.e. the discretization) as long as it is continuous”.

I’m not sure I understand how a mapping couldn’t be continuous. Could you elaborate a bit on this, or give an example of a forbidden mapping?

Does that mean that, for a discretization x_0, …, x_{2^n}, it is valid as long as x_i < x_{i+1} holds? Or is it necessary but not sufficient?

Each state vector maps to a certain range of “x” values.

For instance:
State vector 0 could map to [0.1 to 0.2]
State vector 1 could map to [0.2 to 0.3]
State vector 2 could map to [0.3 to 0.4]

The above would be continuous mapping. Note that in the above example, each bin (e.g. 0.1 to 0.2) is of the same size, but that is not a requirement, just an option.

Here is an example of non-continuous mapping:
State vector 0 maps to [0.1 to 0.2]
State vector 1 maps to [0.2 to 0.3]
State vector 2 maps to [0.35 to 0.5]

The above mapping is not continuous because there is a gap from 0.3 to 0.35 that is not covered.

I hope that helps

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Perfectly clear, thanks!

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