Answer
(a) $f(x)=x$, $f(x)=e^{x}$, $f(x)=x^3+2x-1$
(b) $$f(x)=\left\{_{x,\quad x\in[0,1/2]}^{2x,\quad x\in(1/2,1],}\right.$$
Work Step by Step
(a) The examples are any of the elementary functions that include $[-1,1]$ in their domains. These are $f(x)=x$, $f(x)=e^{x}$, $f(x)=x^3+2x-1$ and so on.
(b) An example can be the function
$$f(x)=\left\{_{x,\quad x\in[0,1/2]}^{2x,\quad x\in(1/2,1],}\right.$$
The left sided limit at $x=1/2$ is
$$\lim_{x\to\frac{1}{2}^-}f(x)=\lim_{x\to\frac{1}{2}^-}x=\frac{1}{2}.$$
The right sided limit at $x=1/2$ is
$$\lim_{x\to\frac{1}{2}^+}f(x)=\lim_{x\to\frac{1}{2}^+}2x=2\cdot\frac{1}{2}=1.$$
They are obviously different so this function is discontinuous at $x=1/2$.
