$F$ is a cumulative distribution function of a random variable $x$ distributed in $[0,1]$ defined as follows:
$$F(x) = ax + b \ \ \ \ \ \ for \ x \leq a$$ $$F(x) = x^2 - x + 1 \ \ \ \ otherwise$$ where $a \in (o,1)$ and $b$ is a real number. Describe the continuity and differentiabilty of $F(x)$ in the interval $(0,1)$ and at $x = a$.
We are actually not taught the probability course, just introduced to the concept of PDF and CDF in our calculus course to help us learn to solve such problems of continuity and differentiability involving them. So I am kind of stumped.
Guide:
For points that are not equal to $a$, clearly they are differentiable.
We have to work on the point $x=a$ separately.
When $x=a$, for it to be continuous, we need
$$\lim_{x \to a^+}F(x) = F(a)$$
Here, you should be able to obtain a linear relation betweeen $a$ and $b$.
To check for differentiablity, check whether you have
$$\lim_{x \to a^{-}}\frac{F(x)-F(a)}{x-a}= \lim_{x \to a^{+}}\frac{F(x)-F(a)}{x-a}$$