Exponential distribution confusion

Question

The PDF of the exponential distribution is $\lambda e^{-\lambda x}$, the CDF is $1-e^{-\lambda x}$.

Assume that, $\lambda=0.8$, $x=0$.

Thus, $P(X \le 0)=1-e^{-0.8 \cdot 0}=0$, and $P(X=0)=0.8e^{-0.8 \cdot 0}=0.8$.

$P(X<0)=P(X \le 0)-P(X=0)=0-0.8=-0.8$.

Where am I wrong?

Answer 1

For continuous random variables, $P(X=x)=0$ for any single point $x$. In particular this means that unlike the discrete case, we don’t have $P(X=x)=f(x)$.

What the density function is used for is that given a set $A$, we have $P(X\in A)=\int_A f(x) dx$.