# Additional Questions [BCB, page 394]

1. Define the continuity of a function at a point. Give with reason, an example of a continuouss function at a point. Is the function $f(x)=\frac{1}{1-x}$ continuous at the point x=1?

2. When a function f(x) is said to be continuous at a point x=a? Discuss the continuity of $f(x)=\begin{cases} x^2+2 & \text {for } x \le 5 \\ 3x+12 & \text{for } x > 5 \end{cases}$ at x=5

3. At what point is the function $f(x)=\frac{x+1}{(x-2)(x-3)}$ (i) discontinuous (ii) continuous ?

4. Doiscuss the continuity of the function f(x) at the point x=0.
$f(x)=\begin{cases} x & \text {for } x > 0 \\ 1 & \text{for } x=0 \\ -x & \text{for } x < 0 \end{cases}$

5. A function f(x) is defind as follows. $f(x)=\begin{cases} 2x+1 & \text {for } x < 1 \\ 2 & \text{for } x=1 \\ 3x & \text{for } x > 0 \end{cases}$
Calculate the left hand limit and right hand limit of f(x) at x=1. Is the function continuous at x=1?

6. What do you understand by the limit of a function? Let a function f(x) is defined by
$f(x)=\begin{cases} 2-x^2 & \text {for } x < 2 \\ 3 & \text{for } x=2 \\ x-4 & \text{for } x > 2 \end{cases}$
Verify that the limit of the function f(x) exists at x=2. Is the function f(x) continuous at x=2? If not why? State how can you make it continuous.

7. A function f(x) is defined as under $f(x)=\begin{cases} \frac{x^2-x-6}{x^2-2x-3} & \text {for } x \ne 3 \\ \frac{5}{3} & \text{for } x =3 \end{cases}$
Prove that f(x) is discontinuous at x=3. Can the definition of f(x) for x=3 be modified so as to make it continuous there?

1. A function f(x) is defined as follows
$f(x)=\begin{cases} \frac{1}{2}+x & \text {for } 0 < x <\frac{1}{2} \\ \frac{1}{2} & \text{for } x =\frac{1}{2} \\ \frac{3}{2}-x & \text{for } \frac{1}{2} < x < 1 \end{cases}$
Show that f(x) has removable discontinuity at $x=\frac{1}{2}$

2. A function f(x) is defined in (0,3) as follows
$f(x)=\begin{cases} x^2 & \text {for } 0 < x <1 \\ x & \text{for } 1 \le x < 2 \\ \frac{1}{4}x^3 & \text{for } 2 \le x < 3 \end{cases}$
Show that f(x) is continuous at x=1 and x=2

3. A function f(x) is defined as follows
$f(x)=\begin{cases} 3+2x & \text {for } -\frac{3}{2} \le x <0 \\ 3-2x & \text{for } 0 \le x < \frac{3}{2} \\ -3-2x & \text{for } x \ge \frac{3}{2} \end{cases}$
Show that f(x) is continuous at x=0 and discontinuous at $x=\frac{3}{2}$

4. A function f(x) is defined as follows
$f(x)=\begin{cases} 1 & \text {for } x >0 \\ 0 & \text{for } x=0 \\ -1 & \text{for } x < 01\end{cases}$
Show that f(x) isdiscontinuous at x=0.

8. In the following, determine the value of the constant so that the given function is continuous at the point mentioned.
1. $f(x)=\begin{cases} kx^2 & \text {for } x \le 2 \\ 3 & \text{for } x > 2\end{cases}$ at x=2

2. $f(x)=\begin{cases}ax+5 & \text {for } x \le 2 \\ x-1 & \text{for } x > 2\end{cases}$ at x=2

3. $f(x)=\begin{cases} 2px+3 & \text {for } x <1 \\ 1-px^2& \text{for } x \ge 1\end{cases}$ at x=1

4. $f(x)=\begin{cases} \frac{x^2-9}{x-3} & \text {for } x \ne 3 \\ k & \text{for } x =3 \end{cases}$ at x=3

9. Let $f(x)=\begin{cases} 2x & \text {for } x <2 \\ 2 & \text{for } x=2 \\ x^2 & \text{for } x >2 \end{cases}$. Show that f(x) has removable discontinuity at x=2.

10. What condition is necessary for a function f(x) to be continuous at the point x=a? In what condition will f(x) be discontinuous at x=a?

11. A function f(x) is defined as $f(x)=\begin{cases} 1 & \text {for } x \ne 0 \\ 2 & \text{for } x =0 \end{cases}$. Find $\displaystyle \lim_{x \to 0} f(x)$ if exists. Is the function continuous at x=0?

12. Find the point of discontinuous of the following functions
1. $f(x)=\frac{x+1}{x-1}$

2. $f(x)=\frac{3x-1}{x^3-5x^2+6x}$