解析学の記事

積分因子による微分方程式の解法1階からn階までのアプローチ

2026年8月11日続きを読む →

$A homogeneous function is not necessarily continuous, as shown by this example. This is the function f {\displaystyle f} defined by f ( x , y ) = x {\displaystyle f(x,y)=x} if x y > 0 {\displaystyle xy>0} and f ( x , y ) = 0 {\displaystyle f(x,y)=0} if x y ≤ 0. {\displaystyle xy\leq 0.} This function is homogeneous of degree 1, that is, f ( s x , s y ) = s f ( x , y ) {\displaystyle f(sx,sy)=sf(x,y)} for any real numbers s , x , y . {\displaystyle s,x,y.} It is discontinuous at y = 0 , x ≠ 0. {\displaystyle y=0,x\neq 0.}$