Proof for the continuity of an recursive function

Recently I am reading one paper “Shaoxiang Chen, M. Lambrecht. XY Band and (s, S) Policies. Research Paper, 1990, No. 9011, ETEW, KU Leuven.” Its proofs for the continuity of a recurisive function is very impressive for me.

The recursive function is:

\[\begin{aligned} &f_n(x)=\min \begin{cases} L(x)+\displaystyle\int f_{n+1}(x-t)\phi(t)dt\\ \min\limits_{x\leq y\leq x+B}K-cx+L(y)+\displaystyle\int f_{n+1}(y-t)\phi(t)dt \end{cases}\\ &f_{N+1}(x)=0 \end{aligned}\]

An assumption: \(L(y)\) is a continuous function on \(x\).

Proof

The proof is by induction. Apparently \(f_{N+1}(x)\) is continuous. Assume \(f_{n+1}(x)\) is continuous, now we prove the continuity of \(f_n(x)\).

• \(V(x)=L(x)+\displaystyle\int f_{n+1}(x-t)\phi(t)dt\) is continuous
• \(U(x)=K-cx+\min\limits_{x\leq y\leq x+B}L(y)+\displaystyle\int f_{n+1}(y-t)\phi(t)dt\) is also continuous. (the proof is very similar to my another blog)
• min function of two continuous functions is also continuous.

Therefore, \(f_n(x)\) is continuous on \(x\).

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