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Continuity of Functions
Examine each of the graphs below. As you scroll up and down, left and right, see if there are holes, vertical asymptotes and jumps in each function's graph.
In checking if a function is continuous, look out for the following.
Check the graph below.
Examining the graph of [math]f\left(x\right)=\frac{1}{x}[/math], is it continuous at all real numbers?
What makes the graph of [math]f\left(x\right)=\frac{1}{x}[/math] not continuous?
What is the point of discontinuity of [math]f\left(x\right)=\frac{1}{x}[/math]?
Continuity of a Function at a Number: Study the 3 Conditions of Continuity.
Observe the graph of f(x) below.
What is [math]f\left(1\right)?[/math]
Is f(x) defined at x=1?
Is Condition 1 satisfied?
What is the limit of f(x) as x approaches 1 from the right?
What is the limit of f(x) as x approaches 1 from the left?
What is the limit of f(x) as x approaches 1?
Is Condition 2 satisfied?
is f(1) EQUAL to the limit of f(x) as x approaches 1?
Is Condition 3 satisfied?
Is f(x) continuous at x = 1?
Study the graph of g(x) below.
Is g(x) continuous at x = -1?
If NO, what condition/s is/are not satisfied?
Study the graph of h(x) below.
What values of x is h(x) not continuous?