A function is continuous in its domain if its graph is a solid unbroken line. All polynomial functions are continuous. Mathematically, (below) a function is defined to be continuous in b if the value of the function is equal to the limit of the function at this point.
Determine the value of the function in x = 2 such that the function is continuous.
A piecewise function
We find one-sided limits
The limit values are equal, so the function has a limit value in x = 2. The function is continuous in x = 2 when determining the value of the function
Let us examine whether the function f is continuous in 4
A function is continuous in 4 if its limit value is equal to the value of the function at this point. A limit value exists if its one-sided limit values are equal.
The value of the function in 4 is determined by the lower expression
The limit value of the function is equal to the value of the function in 4, so the function is continuous at this point.
Graph of f
Let us examine whether the function f is continuous in 5
The one-sided limit values are different, so the function does not have a limit value in 5. Thus, the function is not continuous in 5.
Graph of f
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