Solutions

a) Suppose $f$ is continuous on $[a,b]$. Define

$$g(x)= \begin{cases} f(x), & \text{if } x\in [a,b],\\ f(a), & \text{if } x<a,\\ g(b), & \text{otherwise}. \end{cases}$$

Such a function is obviously continuous for $x\in (-\infty,a)\cup(b,\infty)\cup(a,b)$. For $x=a$,

$$\lim_{x \to a^-}f(x)=\lim_{x \to a^+}f(x)=f(a).$$

Hence, the function $g$ also is continuous at $a$. A similar analysis would show that it also is at $x=b$. We thus have that $g$ is continuous over $\mathbb{R}$.

b) Take $f$ that is continuous on $(a,b)$ but whose limits at either $a$ or $b$ do not exist. In this case, there is no function $g$ that matches the given criteria.