Calculus 2-4

Limits and Continuity

Continuity

• No breaks or interruptions in a function

Discontinuity

• Any breaks in a graph

Continuity at a Point

If the limit does not exist, or if it exists but does not equal then the function is discontinuous at c

Removable Discontinuity

• We can fix the discontinuity by defining the point of discontinuity.

Jump Discontinuity

• If the one side limits exist but are not equal. This is considered bad because it is not an easy fix.

One-Side Continuity

• Left-continuous at

• Right-continuous at

𝐹 (𝑥 )={ 𝑥 𝑓𝑜𝑟 𝑥<13 𝑓𝑜𝑟 1≤ 𝑥≤3𝑥 𝑓𝑜𝑟 𝑥>3

• Continuous at all points except at 1.

• Jump discontinuity at 1

• Right-continuous at 1

Infinite Discontinuity

• If one or both side limits of a function are infinite, then the function is infinite discontinuous at that point.

Basic Laws of Continuity

• If and are continuous at , the following functions are also continuous at

for any constant

if

Continuity of Polynomial and Rational Functions

• Let and be polynomials. Then:

is continuous on the real line.is continuous on its domain (at all values such

that )

Continuity of Some Basic Functions

• is continuous on its domain for n a natural number.

• and are continuous on the real line.• is continuous on the real line (for ). • is continuous for (for ).

Continuity of Composite Functions

• If is continuous at and is continuous at then the composite function is continuous at

Problems

• 2.4 #1-5, 7, 13, 17, 23, 27, 49-53 odd, 57, 59