L5 Continuity - Part 1
Completion requirements
Unit 1B
Limits
Lesson 5: Continuity
In a light switch, current flows continuously and without interruption when the switch is in the ‘on’ position. Once the switch is flipped to the ‘off’ position, the current suddenly stops. Like the current through this circuit, functions can be both
smooth and continuous or they can contain sudden breaks or jumps. In mathematical terms, functions can be either continuous or discontinuous.
A continuous function’s graph has no holes or breaks. Another way to think of a continuous function is one whose graph can be drawn without lifting the pencil from the page.
The video below, The Definition and Properties of Continuous Functions, describes continuity and explains how to determine if a function is continuous.
A continuous function’s graph has no holes or breaks. Another way to think of a continuous function is one whose graph can be drawn without lifting the pencil from the page.
The video below, The Definition and Properties of Continuous Functions, describes continuity and explains how to determine if a function is continuous.
Four different functions are graphed below. Determine if each function is continuous or discontinuous.
This function is continuous because there are no holes or breaks in the graph.
The graph of this function contains no holes or breaks. Although there is a sharp point at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math», it is not necessary to lift the pencil when sketching the graph of this function. As such, this function is continuous.
The graph of this function has a vertical asymptote at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math». This graph cannot be drawn without lifting the pencil from the page. As such, this function is discontinuous at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
There is a break in the graph between the points «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mn»0«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»6«/mn»«/mrow»«/mfenced»«/math». This graph cannot be drawn without lifting the pencil from the page. As such, this function is discontinuous at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»=«/mo»«mn»0«/mn»«/mstyle»«/math».
a.
b.
c.
d.
a.
This function is continuous because there are no holes or breaks in the graph.
b.
The graph of this function contains no holes or breaks. Although there is a sharp point at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mstyle»«/math», it is not necessary to lift the pencil when sketching the graph of this function. As such, this function is continuous.
c.
The graph of this function has a vertical asymptote at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math». This graph cannot be drawn without lifting the pencil from the page. As such, this function is discontinuous at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
d.
There is a break in the graph between the points «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math» and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfenced»«mrow»«mn»0«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»6«/mn»«/mrow»«/mfenced»«/math». This graph cannot be drawn without lifting the pencil from the page. As such, this function is discontinuous at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»=«/mo»«mn»0«/mn»«/mstyle»«/math».