continuity
TRANSCRIPT
CONTINUITYContinuous functions,
discontinuous functions
Continuity at a point a
Continuity at a point ai. exists.
The function is defined at aThe graph of the function contains
Examples of functions not continuous at some x = a
Continuity at a point a
ii. exists
Examples of functions not continuous at some x = a
Continuity at a point a
iii. Example of functions which are not continuous at some point x = a
Continuity at a point a
A function which is not continuous at x = a is discontinuous at that point.
Continuity in an interval
Continuity and Graph
Graphically, a function is continuous in an interval when its graph has no “breaks” or “jumps”.
A function is continuous when one can trace its graph without lifting the pencil from the paper.
Continuity and GraphContinuous functions
Fig 1.8
Continuous Functions
How to demonstrate continuity or discontinuity at a point a
How to demonstrate continuity or discontinuity at a point a
Function is not continuous at x = 2
How to demonstrate continuity or discontinuity at a point a
i. , f(2) does not exist
f is discontinuous at x = 2
ii.
iii.
How to demonstrate continuity or discontinuity at a point a
Removable Discontinuity
If a function has a removable discontinuity at a point a, that discontinuity can be removed by redefining the function to fit continuity, in particular, by making
Removable Discontinuity
If the discontinuities cannot be removed, the discontinuity is called essential discontinuity.
i. f(2) is undefined. The function is discontinuous at 2
ii. exists
Removable Discontinuity
Redefinition of a new function
Removable Discontinuity
Sufficient conditions for continuity Discontinuous at
It is undefined at these values of x
Removable Discontinuity
Sufficient conditions for continuity
Removable Discontinuity
Sufficient conditions for continuity
Essentially discontinuous when . Why?
The discontinuity at x = 0 is removable because
Removable Discontinuity
Redefined function
Theorems on Continuity
Theorems on Continuity
Continuity and Discontintuity
Answer Exercise 3.1, #1 – 8.
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