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Determine whether the function is continuous at
To determine if the function is continuous at we will determine if the three conditions of continuity are satisfied at
Condition 1:
There is no need to proceed further. Condition 2 fails at If any of the conditions of continuity are not satisfied at the function is not continuous at
Determine whether the function is continuous at If not, state the type of discontinuity.
No, the function is not continuous at There exists a removable discontinuity at
Now that we can identify continuous functions, jump discontinuities, and removable discontinuities, we will look at more complex functions to find discontinuities. Here, we will analyze a piecewise function to determine if any real numbers exist where the function is not continuous. A piecewise function may have discontinuities at the boundary points of the function as well as within the functions that make it up.
To determine the real numbers for which a piecewise function composed of polynomial functions is not continuous, recall that polynomial functions themselves are continuous on the set of real numbers. Any discontinuity would be at the boundary points. So we need to explore the three conditions of continuity at the boundary points of the piecewise function.
Given a piecewise function, determine whether it is continuous at the boundary points.
Determine whether the function is discontinuous for any real numbers.
The piecewise function is defined by three functions, which are all polynomial functions, on on and on Polynomial functions are continuous everywhere. Any discontinuities would be at the boundary points, and
At let us check the three conditions of continuity.
Condition 1:
Condition 2: Because a different function defines the output left and right of does
Because ,
Condition 3:
Because all three conditions are satisfied at the function is continuous at
At let us check the three conditions of continuity.
Condition 2: Because a different function defines the output left and right of does
Because , so does not exist.
Because one of the three conditions does not hold at the function is discontinuous at
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