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.

Removable discontinuity 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. 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 𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓𝑜𝑟 𝑥∈𝑅 𝑓 𝑥 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑒𝑥𝑖𝑠𝑡 𝑓𝑜𝑟 𝑥∈𝑅 Condition #1 for a function to be continuous has been met. This curve is continuous. Condition #1 has not been met. This curve is not continuous.

discontinuity Jump 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=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. 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” lim 𝑥→3 𝑓 𝑥 =+∞, lim 𝑥→𝑎 𝑓(𝑥 does not exist for all 𝑥∈𝑅 lim 𝑥→ 𝑎 − 𝑓(𝑥)≠ lim 𝑥→ 𝑎 + 𝑓(𝑥 , lim 𝑥→𝑎 𝑓(𝑥 does not exist all 𝑥∈𝑅 Condition #2 for a curve to be continuous has not been met. This curve is not continuous. Condition #2 for a curve to be continuous has not been met. This curve is not continuous.

Removable 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 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 𝑥→𝑎 𝑓 𝑥 ≠𝑓(𝑎 because b ≠c and so this function is not continuous. 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 𝑥→𝑎 𝑓 𝑥 𝑒𝑥𝑖𝑠𝑡𝑠 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥=𝑎, 𝑥∈𝑅 3. lim 𝑥→𝑎 𝑓 𝑥 =𝑓(𝑎 , 𝑥∈𝑅