# Why do we need continuity for convergence in distribution?

Convergence in Distribution

Let $$P_{n}$$ and $$P$$ be distributions on $$(\mathbb{R},\mathscr{B}(\mathbb{R}))$$ with corresponding cdf's $$F_n(x)=P_{n}((-\infty,x])$$ and $$F(x)=P((-\infty,x])$$ then we say that the distributions $$P_n$$ converge to $$P$$ if

$$\lim\limits_{n\rightarrow\infty}F_n(x)=F(x)$$

for all $$x\in\mathbb{R}$$ in wich $$F$$ is continous.

Why do we need $$F$$ to be continous in $$x$$ here ? What use would this definition have if we would omit the demand for continuity ? I saw an example on wikipedia which didn't really enlighten me.

If we leave out the phrase "...in which $$F$$ is continuous" then we demand more: $$\lim_{n\to\infty}F_n(x)=F(x)$$ should be true every $$x\in\mathbb R$$.
As Kavi makes clear we would not have $$X_n\stackrel{d}{\to}X$$ anymore if $$X_n=\frac1n$$ a.s. and $$X=0$$ a.s. in spite of the fact that $$\lim_{n\to\infty}\frac1n=0$$.
Also $$X_n\stackrel{d}{\to}X$$ if and only if $$\lim_{n\to\infty}\mathbb Ef(X_n)=\mathbb Ef(X)$$ for every continuous and bounded function $$f:\mathbb R\to\mathbb R$$ and this nice equivalence would go lost.
We need the phrase to make certain that the statement "$$X_n\stackrel{d}{\to}X$$" is linked strongly enough with the concept of convergence that is practicized in e.g. analytics.
Even the constant random variables $$1/n$$ will not converge in distribution if you want convergence at every point.