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I am trying to prove the existence of an equilibrium by applying Brouwer's fixed point theorem. In order to invoke this, I of course need my function to be continuous. The only missing step is finding out whether or not this integral is continuous in c on the intervall [0,1]:

$$p_A = \int_{0}^{\frac{1}{2}} \frac{c}{\lambda x + (1-\lambda)} \mathbb{1} (c \in [0, \lambda x + (1-\lambda)]) \mathop{}\mathrm{ d} F(x),$$

where $x$ follows an atomless, continuous distribution F(x) on [0,1], and $\lambda \in [0,1]$.

Can somebody help?

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By DCT it is enough to show that if $c_n \to c$ then $\frac {c_n} {\lambda x+(1-\lambda)} I_{0\leq {c_n} \leq \lambda x+(1-\lambda)} \to \frac c {\lambda x+(1-\lambda)} I_{0\leq c \leq \lambda x+(1-\lambda)} $ for almost all $x$. Note that this convergence holds if $0<c<\lambda x+(1-\lambda)$. Since $c=\lambda x+(1-\lambda)$ for at most one value of $x$ and $F$ is atomless we have completed the proof when $c>0$. The case $c=0$ is much simpler. [Finiteness of $p_A$ is required. If $\lambda <1$ then $\lambda x+(1-\lambda)\geq (1-\lambda)$ which makes the finiteness obvious. If $\lambda =1$ and $c\neq 0$ note that $\int_c^{1/2} \frac 1 x dF(x) <\infty$].

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