Aim: What is the continuity?

Do Now: Given the graph of f(x) Find 1) f(1) 2) lim

𝑥→ 1𝑓 (𝑥)

1 2 3 4

12

3 °.°. °

.𝑓 (1 )=2

lim𝑥→ 1−

𝑓 (𝑥 )=3 lim𝑥→ 1+¿ 𝑓 (𝑥 )=2¿

¿

Left-hand limit ≠ right-hand limit, therefore is not existlim

𝑥→ 1𝑓 (𝑥)

6

2

°f(2) DNE

lim𝑥→ 2−

𝑓 (𝑥 )=6 lim𝑥→ 2+¿ 𝑓 (𝑥 )=6 ¿

¿

f(x) is continuous at x = a iflim𝑥→𝑎

𝑓 (𝑥 )= 𝑓 (𝑎)

f(x) is continuous at x = a means

i) f(a) must be defined

lim𝑥→𝑎

𝑓 (𝑥 )ii) must exist as a finite number

iii)

Continuous function simply means when draw the graph we don’t need to drop our pen or pencil

°.

Jump discontinuity

°.

Removable discontinuity

Example: let 𝑓 (𝑥)={𝑐𝑥+2 ,𝑥<14 −𝑥2 , 𝑥≥ 1

What value of c will f(x) continuous at x = 1?

f(1) is defined

lim𝑥→ 1−

𝑐𝑥+2=𝑐 (1 )+2=𝑐+2

lim𝑥→ 1+¿ 4 −𝑥2=4 −1=3¿

¿

3 = c + 2, c = 1

That means f is continuous at x = 1 only when c = 1

Example: 𝑓 (𝑥)={ 𝑥2− 4𝑥− 2

, 𝑥<2

𝑎𝑥 2−𝑏𝑥+3,2≤ 𝑥<32𝑥−𝑎+𝑏 . 𝑥≥ 3

𝑓 (2 )𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑For what values of a and b will f be continuous on R?

lim𝑥→ 2−

𝑓 (𝑥 )=2+2=4 lim𝑥→ 2+¿ 𝑓 (𝑥 )=4 𝑎−2𝑏+3¿

¿

f(3) is definedlim

𝑥→ 3+¿ 𝑓 (𝑥 )=6 −𝑎+𝑏¿¿lim

𝑥→ 3−𝑓 (𝑥 )=9𝑎− 3𝑏+3

4a – 2b + 3 = 49a – 3b + 3 = 6 – a + b

4a – 2b = 110a – 4b = 3

-8a + 4b = -2 10a – 4b = 3

2a = 1, a = 1/2

) - 2b =1 2 – 2b = 1

2𝑏=1 ,𝑏=12