# Continuity of integral of indicator function

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?

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 $$0. Since $$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$$].