Plot a discontinuous function in R without connecting a "jump"

You could write a little wrapper function which finds discontinuities in the given function and plots them as separate groups:

plot_fun <- function(fun, from = 0, to = 1, by = 0.001) {
  
  x <- seq(from, to, by)
  groups <- cut(x, c(-Inf, x[which(abs(diff(fun(x))) > 0.1)], Inf))
  df <- data.frame(x, groups, y = fun(x))
  
  ggplot(df, aes(x, y, group = groups)) +
    geom_line()
}

This allows

plot_fun(f)

plot_fun(floor, 0, 10)

This answer is based on Allan Cameron's answer, but depicts the jump using open and closed circles. Whether the function is right or left continuous is controlled by an argument.

library("ggplot2")
plot_fun <- function(fun, from = 0, to = 1, by = 0.001, right_continuous = TRUE) {

  x <- seq(from, to, by)
  tol_vertical <- 0.1
  y <- fun(x)
  idx_break <- which(abs(diff(y)) > tol_vertical)
  x_break <- x[idx_break]
  y_break_l <- y[idx_break]
  y_break_r <- y[idx_break + 1]

  groups <- cut(x, c(-Inf, x_break, Inf))
  df <- data.frame(x, groups, y = fun(x))

  plot_ <- ggplot(df, aes(x, y, group = groups)) +
            geom_line()

  # add open and closed points showing jump
  dataf_l <- data.frame(x = x_break, y = y_break_l)
  dataf_r <- data.frame(x = x_break, y = y_break_r)
  shape_open_circle <- 1
  # this is the default of shape, but might as well specify.
  shape_closed_circle <- 19
  shape_size <- 4
  if (right_continuous) {
    shape_l <- shape_open_circle
    shape_r <- shape_closed_circle
  } else {
    shape_l <- shape_closed_circle
    shape_r <- shape_open_circle
  }

  plot_ <- plot_ +
    geom_point(data = dataf_l, aes(x = x, y = y), group = NA, shape = shape_l, size = shape_size) +
    geom_point(data = dataf_r, aes(x = x, y = y), group = NA, shape = shape_r, size = shape_size)
  return(plot_)
}

Here's the OP's original example:

f <- function(x){
  (x < .5) * (x) + (x >= .5) * (x + 1)
}
plot_fun(f)

Here's Allan's additional example using floor, which shows multiple discontinuities:

plot_fun(floor, from = 0, to = 10)

And here's an example showing that the function does not need to be piecewise linear:

f_curved <- function(x) ifelse(x > 0, yes = 0.5*(2-exp(-x)), no = 0)
plot_fun(f_curved, from = -1, to = 5)

You can insert everything inside an ifelse:

f <- function(x){
  ifelse(x==0.5, 
         NA, 
         (x < .5) * (x) + (x >= .5) * (x + 1))
}
ggplot()+ 
  geom_function(fun = f)