Continuous Random Variable I know that for a continuous random variable the value of $P(X=x) = 0$ for all values of $x$. But when I take 2 points on the graph of a continuous random variable's CDF then the graph comes not to be a straight line.
But I expect it to be constant at all times because for all $x$, $P(X=x) =0$ and thus the cumulative sum should not increase. Can anyone tell where I am wrong?

Comment with figure:
Roughly speaking, you might think of it this way: You can have two values of the standard normal density function $\varphi(\cdot)$ at $z = 1.00, 1.01,$ with $\varphi(1.00), \varphi(1.01),$ with 'heights' $0.2419707, 0.2395511,$ respectively--both about $0.24.$
In R, where dnorm is standard normal density $\varphi$ and pnorm is standard normal CDF $\Phi:$
dnorm(c(1,1.01))
[1] 0.2419707 0.2395511
diff(pnorm(c(1,1.01)))
[1] 0.002407609

Then $P(1.00 < Z <1,01)=0.002407609$ $\approx 0.0024 = 0.24(0.01)$ is the area of the almost-rectangular region under $\varphi(\cdot)$ between $1.00$ and $1.01.$
diff(pnorm(c(1,1.01)))
[1] 0.002407609


However, $P(Z = 1.00) = 0$ would just be the "area" or a line.
R code for figure:
hdr = "Standard Normal Density Function"
curve(dnorm(x), 0, 1.2, ylim=c(0,0.4), ylab="PDF", 
      xlab="z", xaxs="i", lwd=2, main=hdr)
 abline(v=c(1,1.01), col="blue")
 abline(h=0.24, col="blue", lty="dotted")
 abline(h=0, col="green2")

