Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Functional programming, design and analysis of algorithms / 5 / d / i / Solution

If coordinate $j$ is not an essential variable of a function, changing that coordinate alone never changes the output, for any values of the other coordinates. Take two inputs with the same $i$-th coordinate. Change their other coordinates one at a time to transform the first input into the second. Every such change is inessential, so all intermediate outputs are equal.

Define $f_i(t)=f(1,\dots ,1,t,1,\dots ,1)$, with $t$ in position $i$. The preceding argument gives $f(x_1,\dots ,x_n)=f_i(x_i)$. Since $i$ is essential, $f_i$ is nonconstant. The conclusion depends on the full “for every context” negation of essentiality, not merely on testing a coordinate at one selected input.