Continuity of A Function

A function f(x) is continuous at x = c if and only if all three of the following tests hold:

i. f c exists

x c

ii. lim f x exists

x c

iii. lim f x f c

f(x) is right continuous at x = -5

f(x) is continuous at x = -4

f(x) has infinite discontinuity at x = -3 [i, iii]

f(x) has point discontinuity at x = -2 [i, iii]

f(x) has infinite discontinuity at x = -1 [i, ii, iii]

f(x) is continuous at x = 0

At x = 1

At x = 2

At x = 3

At x = 4

At x = 5

Point Discontinuity [i, iii]

Jump Discontinuity [i, ii, iii]

Continuous

Continuous

Point Discontinuity [i, (ii), iii]

2f x x 2x 1

continuous

2

1f x

x 1

continuous

2

xf x

x x

x

x x 1

pt. discontinuity at x = 0

inf. discontinuity at x = 1

2

x 3f x

x 9

x 3

x 3 x 3

pt. discontinuity at x = 3

inf. discontinuity at x = -3

2

2x 3 x 1f x

x x 1

2 1 3 1

21 1

continuous

1

x 1 x 2f x 2

3 x x 2

12 1 2

2

3 2 1

jump discontinuity at x = 2

Find the value of a which makes the function below continuous

3

2

x x 2f x

ax x 2

3

2

3

2

x x 2f x

ax x

2 8

2 4a2 a

4a 8 a 2

Find (a, b) which makes the function below continuous

2 x 1

f x ax b 1 x 3

2 x 3

As we approach x = -1 2 = -a + b

As we approach x = 3 -2 = 3a + b

4 4aa 1 b 1

1,1

Consider the function sinx

x 0f x x

k x 0

Find the value of k which makes f(x) continuous at x = 0

Sincex 0

sinxlim 1

x , if k =1, the hole is filled.

Continuity of a Composite Function

If f is continuous at c, and if g is continuous at f(c )

then g f or g f x is continuous at c.

2f x x 1 g x x

Since f is continuous at all c, and since g is continuous at all f(c ),

g f is continuous

1f x x 1 g x

x

Since f is continuous at all c, and since g is discontinuous at all f(c ) < 0,

g f is discontinuous at x – 1 < 0

g f is discontinuous at x < 1