- #1
Eclair_de_XII
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- 91
Homework Statement
"Show that if ##P(X=c)>0##, for some ##c\in \mathbb{R}##, then the distribution function ##F_X(x)=P(X\leq x)## is discontinuous at ##c##. Is the converse true?"
Homework Equations
Continuity of a distribution function: ##\lim_{\epsilon \rightarrow 0}F_X(x+\epsilon)=F_X(x)##
The Attempt at a Solution
I'm thinking that the graph of ##F_X(x)## jumps up at ##x=c##. So far, I have:
##\lim_{\epsilon \rightarrow 0}F_X(c+ \epsilon)\\
=\lim_{\epsilon \rightarrow 0} P(X\leq c + \epsilon)\\
=\lim_{\epsilon \rightarrow 0}P(X<c+\epsilon)+\lim_{\epsilon \rightarrow 0}P(X=c+\epsilon)\\
=P(X\leq c)+P(X=c)>P(X\leq c)##
I'm not sure how to justify the step made in the third line, stating that: ##\lim_{\epsilon \rightarrow 0}P(X<c+\epsilon)=P(X\leq c)##.