Clarification on Definition of a continuous random variable

Question

My confusion lies with the statement : "The student will note that $F'(x)$ does not exist at $x=a$ and $x=b$. Why is this true? As far as I understand $F'(a) = F'(b) = f(a) = f(b) = \frac {1}{b-a} $. What am I getting wrong?

Answer 1

Actually you have that the right and the left derivative exist at $a$ $$f'_-(a)=\lim_{x\to a^-}\frac{f(x)-f(a)}{x-a}=\lim_{x\to a^-}\frac{0-0}{x-a}=0,\\f'_+(a)=\lim_{x\to a^+}\frac{f(x)-f(a)}{x-a}=\lim_{x\to a^+}\frac{\frac{x-a}{b-a}-0}{x-a}=\frac{1}{b-a}.$$ Since they are different, the derivative at $a$ does not exist. A similar argument works at $b$.