A future-directed <causal curve> from the origin has $|dx/dt|\le c$. Integrating this inequality gives the <causal future>
$$
\boxed{J^+(0)=\{(t,x):ct\ge|x|\}.}
$$
The straight timelike or null segment proves sufficiency. This is the blue region above the two null rays in the upper-left <spacetime diagram> of part <1/c>.

Solution

Past exam of the symmetry and relativity course of the University of Oxford / 2019 / 1 / c / i / Solution

 Solution (source code)