Chapter 1.4, Problem 1C

(a)

To determine

Whether given statement is true or false.

‘If f is both left and right continuous at $x=c$ , then f is continuous.’

(b)

To determine

Whether given statement is true or false.

If function is continuous on an open interval $(0,5)$ then f is continuous at every point in $(0,5)$

(c)

To determine

Whether given statement is true or false.

If function is continuous on the closed interval $[0,5]$ then f is continuous at every point in $[0,5]$

(d)

To determine

Whether given statement is true or false.

‘If f is continuous on the interval $(2,4)$ then f must have a maximum and minimum value on $(2,4)$ ’

(e)

To determine

Whether given statement is true or false.

‘If $f(3)=-5$ and $f(9)=-2$ , then there must be a value ‘c’ at which $f(c)=-3$ ’

(f)

To determine

Whether given statement is true or false.

‘If f is continuous everywhere and if $f(-2)=3$ and $f(1)=2$ then $f(x)$ must have a root somewhere in $(-2,1)$ ’

(g)

To determine

Whether given statement is true or false.

‘If f is continuous everywhere and if $f(0)=-2$ and $f(4)=3$ then $f(x)$ must have a root somewhere in $(0,4)$ ’

(h)

To determine

Whether given statement is true or false.

‘If $f(0)=f(6)=0$ and $f(2)>0$ then $f(x)$ is positive on the entire interval $(0,6)$ ’