Continuity and Differentiability

9. Given, f (x) = $\{\begin{array}{cc}\hfill x+1,\pi x\ge 1\hfill & \hfill x^2+1, \pi x<1.\hfill \end{array}$

For x = c < 1,

$\underset{x\to c}{lim}$ f (x) = $\underset{x\to c}{lim}$ x2 + 1 = c2 + 1

∴ $\underset{x\to c}{lim}$ f (x) = f (c)

So f is continuous at x = c < 1.

For x = c > 1,

F (c) = c + 1

$\underset{x\to c}{lim}$ f (x) = $\underset{x\to c}{lim}$ x + 1 = c + 1

∴ $\underset{x\to c}{lim}$ f (x) = f (c)

So, f is continuous at x = c > 1.

For x = c = 1, + (1) = 1 + 1 = 2

L.H.L. = $\underset{x\to 1^{-}}{lim}$ f (x) = $\underset{x\to 1^{-}}{lim}$ x2 + 1 = 12 + 1 = 2.

R.H.L. = $\underset{x\to 1^{+}}{lim}$ f (x) = $\underset{x\to 1^{+}}{lim}$ x + 1 = .1 + 1 = 2

∴ L.H.L = R.H.L. = f (1)

So, f is continuous at x = 1.Hence f has no point of discontinuity.

8. Given, f(x) = $fxxx$

For x = c < 0,

f(c) = 1

$\underset{x\to 0}{lim}$ f(x) = $\underset{x\to 0}{lim}$ 1 = 1

∴f(c) = $\underset{x\to 0}{lim}$ f (x)

f is continuous at x $\left|<\right|$ 0.

For x = c > 0,

F (c) = 1

$\underset{x\to c}{lim}$ f(x) = $\underset{x\to c}{lim}$ $\cancel{=}$ = 1.

∴f(c) = $\underset{x\to c}{lim}$ f(x)

f is continuous at x > 0.

For x = c 0.

L.H.L. = $\underset{x\to 0^{-}}{lim}$ f(x) = $\underset{x\to 0^{-}}{lim}$ ( 1) = 1

R.H.L. $\underset{x\to 0^{+}}{lim}$ f(x) = $\underset{x\to 0^{+}}{lim}$ 1 = 1

∴ L.H.L. $\cancel{=}$ R.H.L.

is now continuous at x = 0, point of discontinuity of f is at x = 0.

6. Given f(x) = $\{\begin{array}{cc}\hfill 2x+3 if x\le 2\hfill & \hfill 2x-3 if x>2.\hfill \end{array}$

For x = c < 2,

F (c) = 2c + 3

$\underset{x\to c}{lim}$ f(x) = $\underset{x\to c}{lim}$ 2x + 3 = 2c + 3

∴ $\underset{x\to c}{lim}$ f (x) = f(c)

So f is continuous at x $\left|<\right|$ 2.

For x = c > 2.

F (c) = 2c 3

$\underset{x\to c}{lim}$ f(x) = $\underset{x\to c}{lim}$ 2x 3 = 2c 3

∴ $\underset{x\to c}{lim}$ f(x) = f(c)

So f is continuous at x $\left|>\right|$ 2.

For x = c = 2,

L.H.L. = $\underset{x\to 2^{-}}{lim}$ f(x) = $\underset{x\to 2^{-}}{lim}$ .2x + 3 = 2. 2 + 3 = 4 + 3 = 7.

R.H.L. = $\underset{x\to 2^{+}}{lim}$ f(x) = $\underset{x\to 2^{+}}{lim}$ 2x 3 = 2. 2 3 = 4 3 = 1.

∴ LHL $\cancel{=}$ RHL

∴ f is not continuous at x = 2.i e, point of discontinuity

5. Given, f (a) = $\{\begin{array}{ll}x, if x\le 1\hfill & 5, if x>1.\hfill \end{array}$

At x = 0,

(0) = 0

$\underset{x\to 0}{lim}$ f (x) = $\underset{x\to 0}{lim}$ x = 0

∴ $\underset{x\to 0}{lim}$ f (x) = f (0)

So, f is continuous at x = 0.

At x = 1,

Left hand limit,

L.H.L = $\underset{x\to 1}{lim}$ f (x) = $\underset{x\to 1}{lim}$ x = 1.

Right hand limit,

R. H. L. = $\underset{x\to 1^{+}}{lim}$ f (x) = $\underset{x\to 1^{+}}{lim}$ 5 = 5.

L.H.L. $\cancel{=}$ R.H.L.

So, f is not continuous at x = 1.

At x = 2,

f (2) = 5.

$\underset{x\to 1}{lim}$ f (x) = $\underset{x\to 2}{lim}$ 5 = 5

$\underset{lim\to 2}{limf}$  (x) = f (2)

So f is continuous at x = 2.

Find all points of discontinuity of f, where f is defined by

4. Given, f (x) = x n > n = positive.

At x = 2,

(x) = n.

$\underset{x\to n}{lim}$ f (x) = $\underset{x\to n}{lim}$ x n = n

∴ $\underset{x\to n}{lim}$ f (x) = f (x)

So f is continuous at x = n.

2. Given, f (x) = 2x2 1

At x = 3

Lim f (x) = $\frac{dy}{dx}=\frac{-\left(3x^2+2xy+y^2\right)}{\left(x^2+2xy+3y^2\right).}$ 2 (3)2 1 = 18 1 = 17.

So, f is continuous at x = 3.

1. Given, f (x) = 5x 3

At x = 0,  $\underset{x\to 0}{lim}f(x)=\underset{x\to 0}{lim}$ 5x 3 = 5 0 3 = 3.

So f is continuous at x = 1.

At x = 3,  $\pi +h$ 5x 3 = 5 ( 3) 3 = 15 3

= 18.

So f is continuous at x = 3.

At x = 5,  $\forall x\in ?$ .5x 3 = 5.5 3 = 25 3 = 22.

So, f is continuous at x = 5.