📝 题目
6. 1.20 设 $f\left( {x,y}\right)$ 在 ${x}^{2} + {y}^{2} \leq {R}^{2}$ 上可积, $0 < h < R$ ,令
$$ F\left( {\xi ,\eta }\right) = {\iint }_{{\left( x - \xi \right) }^{2} + {\left( y - \eta \right) }^{2} \leq {h}^{2}}f\left( {x,y}\right) \mathrm{d}x\mathrm{\;d}y. $$
证明: $F\left( {\xi ,\eta }\right)$ 在 ${\xi }^{2} + {\eta }^{2} < {\left( R - h\right) }^{2}$ 上连续.
💡 答案与解析
6.1.20 证 $\left| {F\left( {\xi + {\Delta \xi },\eta + {\Delta \eta }}\right) - F\left( {\xi ,\eta }\right) }\right| \leq {2M\pi }\left\lbrack {{h}^{2} - {\left( h - \delta \right) }^{2}}\right\rbrack$ ,其中
$$ M = \mathop{\sup }\limits_{{{x}^{2} + {y}^{2} \leq {R}^{2}}}\left| {f\left( {x,y}\right) }\right| ,\;\delta = \sqrt{\Delta {\xi }^{2} + \Delta {\eta }^{2}}. $$