Continuous or not?
If a curve is not continuous, then it is discontinuous.
Decide whether each of the following is
continuous or not.
Explain your reasoning.
Continuous function because,
• in theory, you could trace along the curve from left to
right without once having to lift your pencil.
• There is a y-value for each 𝑥 ∈ 𝑅
This is not a continuous function because
• if you were to attempt to trace the curve with your
pencil from left to right, you would have to lift your
pencil at “x=a” and then reposition your pencil on
the paper, and then also not be able to draw to
the right of “x = E”.
𝑓 𝑎 𝑎𝑛𝑑 𝑓 𝐹 𝑑𝑜 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡
• y-values do not exist for any x-value chosen.
𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓𝑜𝑟 𝑥 ∈ 𝑅
Condition #1 for a function to be continuous has been met.
This curve is continuous.
𝑓 𝑥 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑓𝑜𝑟 𝑥 ∈ 𝑅
Condition #1 has not been met.
This curve is not continuous.
Infinite discontinuity
This function is not continuous
• If you were to attempt to draw this function
from left to right, you would have to lift your
pencil at “x=a” reposition your pencil’s height
and then continue drawing the function out
to the right.
• The limit of f(x) from the left does not
approach the same value as the limit of f(x)
from the right as “x approaches a”
This function is not continuous
• If you were to attempt to draw this function from left to
right, you would have to lift your pencil at “x=3” and
reposition your pencil on the other side of the vertical
asymptote to continue drawing the function to the right.
• The limits of f(x) from the left and the right approach infinity
which is not a specific value and therefor the limit of f(x) as
“x approaches 3” does not exist.
lim 𝑓 𝑥 = +∞,
𝑥→3
lim− 𝑓(𝑥) ≠ lim+𝑓(𝑥 ) ,
𝑥→𝑎
𝑥→𝑎
lim 𝑓(𝑥 ) does not exist all 𝑥 ∈ 𝑅
𝑥→𝑎
Condition #2 for a curve to be continuous has not been met.
This curve is not continuous.
lim 𝑓(𝑥 ) does not exist for all 𝑥 ∈ 𝑅
𝑥→𝑎
Condition #2 for a curve to be continuous has not been met.
This curve is not continuous.
This function is not continuous
• If you were to attempt to draw this function from left to
right, you would have to lift your pencil at “x=a” reposition
your pencil’s height to plot (a, f(a) ) and lift your pencil again
and change its height again before being able to then
continue drawing the function out to the right.
• The limit of f(x) as “x approaches a” exists, but this limit is
not equal to the at an instance value “f(a)”.
lim 𝑓 𝑥 ≠ 𝑓(𝑎) because b ≠c and so this function is not continuous.
𝑥→𝑎
Condition #3 has not been met.
Three algebraic conditions
for a function to be continuous.
All three conditions must be satisfied or met
in order for the function to be continuous.
1.
𝑓 𝑎 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 = 𝑎,
𝑥∈𝑅
2. lim 𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 = 𝑎,
𝑥∈𝑅
𝑥→𝑎
3.
lim 𝑓 𝑥 = 𝑓(𝑎) ,
𝑥→𝑎
𝑥∈𝑅