Continuous functions

Exercise

1.

Find the following function $f(x)$ is continuous at $x = 2$.

$$f(x) = \left\{\begin{array}{cl} \frac{x^2 - x - 2}{x - 2}, & x \neq 2\\ 3, & x = 2 \end{array} \right.$$

2.

Show that $f(x) = \sqrt{x}$ is continuous on $[0,\infty)$．

3.

Find the max and min of the following functions:

(a)

$${f(x) = x^2 - 3x + 1, \ x \in [-2,1]}$$

(b)

$${f(x) = \frac{1}{x}, \ x \in (0,1]}$$

(c)

$${f(x) = x^2 - ax, \ x \in [0,2]}$$

4.

Prove that $${2\sin{x} - x = 0}$$ has the real solution in $${(\frac{\pi}{2},\pi)}$$.