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[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''³}} with an inflection point at (0,0), which is also a [[stationary point]].|链接=Special:FilePath/X_cubed_plot.svg]]

[[Image:x cubed plot.svg|thumb|Plot of {{math|1=''y'' = ''x''³}} with an inflection point at (0,0), which is also a [[stationary point]].|链接=Special:FilePath/X_cubed_plot.svg]]

Plot of with an inflection point at (0,0), which is also a [[stationary point.]]

Plot of {{math|1=''y'' = ''x''³}} with an inflection point at (0,0), which is also a [[stationary point.]]

拐点在(0,0) ，也是一个[[驻点]]。

{{math|1=''y'' = ''x''³}} 的拐点是(0,0) ，也是一个[[驻点]]。

{{Cubic graph special points.svg}}

{{Cubic graph special points.svg}}

驻点拐点例子：{{math|(0, 0)}} 为函数 {{math|''y'' {{=}} ''x''³}} 的驻点 。切线为 {{mvar|x}} 轴，在{{math|(0, 0)}}与函数相切。

驻点拐点例子：{{math|(0, 0)}} 为函数 {{math|''y'' {{=}} ''x''³}} 的驻点 。切线为 {{mvar|x}} 轴，在{{math|(0, 0)}}与函数相切。

An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''³ + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.

An example of a non-stationary point of inflection is the point {{math|(0, 0)}} on the graph of {{math|''y'' {{=}} ''x''³ + ''ax''}}, for any nonzero {{mvar|a}}. The tangent at the origin is the line {{math|''y'' {{=}} ''ax''}}, which cuts the graph at this point.

An example of a non-stationary point of inflection is the point  on the graph of  x³ + ax}}, for any nonzero . The tangent at the origin is the line  ax}}, which cuts the graph at this point.

非驻点拐点例子：{{math|(0, 0)}}为函数 {{math|''y'' {{=}} ''x''³ + ''ax''}} 的非驻点（{{mvar|a}}为任意非零常数）。切线为 {{math|''y'' {{=}} ''ax''}}，在{{math|(0, 0)}}与函数相切。

==Functions with discontinuities==

==Functions with discontinuities==

Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative {{mvar|x}} and convex for positive {{mvar|x}}, but it has no points of inflection because 0 is not in the domain of the function.

Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative {{mvar|x}} and convex for positive {{mvar|x}}, but it has no points of inflection because 0 is not in the domain of the function.

Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative  and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.

Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function <math>x\mapsto \frac1x</math> is concave for negative  and convex for positive , but it has no points of inflection because 0 is not in the domain of the function.

有些函数在没有拐点的情况下也可改变凹度。他们可以通过改变垂直渐近线或非连续性来实现。例如，函数 < math > x mapsto frac1x </math > 在x为负的时候显凹性，在x为正的时候显凸性。但这个函数不具有拐点，因为0不在其定义域内。

有些函数虽然没有拐点但也可改变凹度。这些函数可以通过改变垂直渐近线或非连续性来改变凹度。例如，函数 < math > x mapsto frac1x <nowiki></math ></nowiki> 在函数值为负时显凹性，在函数值为正时显凸性。但这个函数不具有拐点，因为0不在其定义域内。