Is (-1)^x a discontinuous function?

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The function y=(-1)^x is discontinuous because it oscillates between positive and negative values depending on whether x is an even or odd integer. This creates ambiguity for non-integer values, such as irrational numbers like pi, where the even or odd classification does not apply. The function is only defined for real numbers when abs(x) > 0, leading to undefined values for certain inputs. For example, when x=0.5, the function's output is related to the square root of -1, which is an imaginary number. Thus, the discussion highlights the complexities of defining y=(-1)^x across the real number line.
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So I was thinking about the equation y=(-1)x. This equation will jump back and fourth across the x-axis depending on weather x is even or not. This must mean that the function is discontinuous. Figuring out if the value is positive or negative is straight forward for ration numbers but how can we tell if say (-1)^pi is negative or positive since we don't know if pi is even or odd.
 
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It's not that simple. Remember x is a continuous variable, and the quality of even or odd applies only to integers. There are no even or odd rational or irrational numbers.

The equation in the OP is defined for real numbers only when abs(x) > 0. Do you know what y equals when x equals 1/2?
 
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I never thought of what it would be be when x=0.5. So I guess it can be positive negative or undefined.
 
Don't you know what it means to raise a number to a power of 0.5? Does square root come to mind?

What is the square root of -1?
 

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