Uniroot error when estimating coverage probability for an unspecified distribution

Question

I am trying to compute the coverage probability for an unspecified distribution. I have copied my r codes below and the corresponding error message.

Firstly, I defined the CDF as below.

CD_theta <- function(x, p, n) {
        1  -  pbinom (x, size = n, prob = p) + 1/2 * dbinom(x, size = n, prob = p)

Then used the following command to estimate the coverage probability.

N <- 10000
n <- 10
p <- 0.5
set.seed(1)

x <- rbinom(N, prob=p, size=n)

covered <- numeric()
cilower <- numeric() 
ciupper <- numeric() 
for (i in 1:N) {
  CD_theta <- function(x, p, n) {
    1  -  pbinom (x, size = n, prob = p) + 1/2 * dbinom(x, size = n, prob = p)
  }
   cilower <- uniroot(function(p) CD_theta(x[i], p, n) - 0.025, c(0, 1))$root
   ciupper <- uniroot(function(p) CD_theta(x[i], p, n) - 0.975, c(0, 1))$root
   covered[i] <- (cilower < p) & (p < ciupper) 
   }
mean(covered) 

Error message and the output are below.

Error in uniroot(function(p) CD_theta(x[i], p, n) - 0.975, c(0, 1)) : f() values at end points not of opposite sign

mean(covered) [1] 0.9820282

Am I doing any mistake in coding? How can I prevent this?