形如f(x)=x^n,n为偶数的函数称为偶函数；形如f(x)=x^n,n为奇数的函数称为奇函数。另一种看法：如果光滑，偶函数的Taylor展开中只有偶次数项，奇函数的Taylor展开中只有奇次数项.简单来说，幂函数的幂为偶数则是偶函数，这时函数图像关于Y轴对称；奇函数同理。后来被推广到所有函数的图像性质上。
偶函数的最早是由欧拉命名的，拉丁文为 Functiones pares 最早可见于1727年欧拉提交给Petersburg Academy的"Problematis traiectoriarum reciprocarum solutio," （原文不是英语，我看不懂，有兴趣自己百度吧）
英语文献记载：In the first place are noted functions, which I call even, of which there is this property, that they remain unchanged if in place of x is put ?x. The exponents of such functions are even numbers, or fractions with numerators that are even numbers and denominators that are odd. Then, functions composed of functions of this kind, by addition of subtraction or multiplication or division or by elevation to any power will also be even.Secondly, I observe odd functions, which produce their own negatives, if in place of x goes ?x. Functions of this kind are x itself, x^3, x^5, etc. all powers the exponents of which are odd numbers, of fractions the numerators and denominators of which are odd numbers. Also, functions which are formed by addition or subtraction of such powers, or their elevation to odd powers, are odd functions. 以上。
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