To check for continuity at the edges of each piece, you need to consider the limit as [tex]x[/tex] approaches the edges. For example,
[tex]g(x)=\begin{cases}2x+5&\text{for }x\le-3\\x^2-10&\text{for }x>-3\end{cases}[/tex]
has two pieces, [tex]2x+5[/tex] and [tex]x^2-10[/tex], both of which are continuous by themselves on the provided intervals. In order for [tex]g[/tex] to be continuous everywhere, we need to have
[tex]\displaystyle\lim_{x\to-3^-}g(x)=\lim_{x\to-3^+}g(x)=g(-3)[/tex]
By definition of [tex]g[/tex], we have [tex]g(-3)=2(-3)+5=-1[/tex], and the limits are
[tex]\displaystyle\lim_{x\to-3^-}g(x)=\lim_{x\to-3}(2x+5)=-1[/tex]
[tex]\displaystyle\lim_{x\to-3^+}g(x)=\lim_{x\to-3}(x^2-10)=-1[/tex]
The limits match, so [tex]g[/tex] is continuous.
For the others: Each of the individual pieces of [tex]f,h[/tex] are continuous functions on their domains, so you just need to check the value of each piece at the edge of each subinterval.
