discontinuous是什么意思，discontinuous翻译

discontinuous是什么意思，discontinuous翻译

discontinuous

In the field of mathematics, the term "discontinuous" refers to a function or sequence that does not have a continuous behavior. A continuous function is one in which the graph can be drawn without lifting the pen from the paper, meaning that there are no sudden jumps, breaks, or holes in the graph. On the other hand, a discontinuous function has one or more points where the function is not continuous, resulting in a graph with jumps, breaks, or holes.

There are different types of discontinuities, including removable discontinuities, jump discontinuities, and infinite discontinuities. A removable discontinuity occurs when there is a hole in the graph of a function, which can be filled by redefining the function at that point. A jump discontinuity occurs when there is a sudden jump in the function's value at a specific point, resulting in a vertical gap in the graph. An infinite discontinuity occurs when the function approaches infinity as it approaches a certain point, resulting in a vertical asymptote in the graph.

One common example of a discontinuous function is the absolute value function, f(x) = |x|. This function is discontinuous at x = 0, as the function changes from negative to positive values at that point, creating a jump discontinuity. The graph of the absolute value function shows a sharp corner at x = 0, representing this discontinuity.

Another example of a discontinuous function is the rational function f(x) = 1/x. This function is discontinuous at x = 0, as the function approaches infinity as x approaches 0. The graph of the rational function shows a vertical asymptote at x = 0, representing this discontinuity.

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