# X 轴和 Y 轴分量上指定点的偏导数存在且在该点处连续与该点可微之间没有任何必然联系

## 二、解析

### 方法一：逻辑分析法

set term wxt size 800,600  # 设置绘图窗口尺寸
set xlabel "{/:Bold=15 x}" textcolor rgb "orange"  # 设置x轴标签
set ylabel "{/:Bold=15 y}" textcolor rgb "orange"  # 设置y轴标签
set zlabel "{/:Bold=15 z}" textcolor rgb "orange"  # 设置z轴标签
set view 60, 30           # 设置视角
set hidden3d              # 启用3D隐藏线算法
set style line 1 lc rgb "green" lw 2  # 设置函数图象线条样式
set border lc rgb "orange" lw 2  # 设置边框样式
set tics textcolor rgb "orange"  # 设置刻度线标签颜色
set key off  # 关闭图例
set style fill transparent solid 0.5  # 设置背景透明度
set zeroaxis lt -1 lc rgb "orange"  # 设置零点辅助线样式

# 定义函数
f(x, y) = x**2 + y**2

# 设置x轴和y轴范围
set xrange[-5:5]
set yrange[-5:5]

# 绘制图形
splot f(x, y) with lines ls 1

set term wxt size 800,600  # 设置绘图窗口尺寸
set xlabel "{/:Bold=15 x}" textcolor rgb "orange"  # 设置x轴标签
set ylabel "{/:Bold=15 z}" textcolor rgb "orange"  # 设置y轴标签
set view 0, 0            # 设置视角
set style line 1 lc rgb "green" lw 2  # 设置函数图象线条样式
set border lc rgb "orange" lw 2  # 设置边框样式
set tics textcolor rgb "orange"  # 设置刻度线标签颜色
set key off  # 关闭图例
set style fill transparent solid 0.5  # 设置背景透明度

# 定义函数
f(x) = x**2

# 绘制图形
set xrange[-5:5]
set yrange[0:25]
plot f(x) with lines ls 1

$$\lim \limits_{x \rightarrow 0} f_{x}^{\prime}(x, 0) \neq f_{x}^{\prime}(0,0)$$

$$\lim \limits_{y \rightarrow 0} f_{y}^{\prime}(0, y) \neq f_{y}^{\prime}(0,0)$$

### 方法二：反例法

#### 反例一

$$g(x, y)=\left\{\begin{array}{l} x y, & x y \neq 0 \\ 1, & x y=0 \end{array}\right.$$

$$g_{x}^{\prime}(x, y)=x \quad g_{y}^{\prime}(x, y)=y \Rightarrow$$

$$\lim \limits_{\substack{x \rightarrow 0 \\ y=0}} g_{x}^{\prime}(x, 0)=0=g_{x}^{\prime}(0,0)$$

$$\lim \limits_{\substack{x=0 \\ y \rightarrow 0}} g_{y}^{\prime}(0, y)=0=g_{y}^{\prime}(0,0)$$

set terminal wxt enhanced font "arial,10" fontscale 1.0 size 600, 600

set border linewidth 1.5 linecolor rgb '#FFA500'  # 橙色
set tics textcolor rgb '#FFA500'  # 橙色
set style line 1 lc rgb '#008000' lt 1 lw 2  # 绿色

# 定义分段函数
z(x,y) = x*y != 0 ? x*y : 1

# 设置绘图范围和密度
set xrange [-30:30]
set yrange [-30:30]
set zrange [-30:30]
set isosamples 80, 80  # 设置绘图密度

# 设置视角
set view 60, 50, 1.0, 1.0

# 绘制 z(x, y)
splot z(x, y) with lines linestyle 1

# 添加过 x=0 点且平行于 y 轴的坐标轴
set arrow from 0, graph 0, first 0 to 0, graph 1, first 0 nohead linecolor rgb '#FFA500'

pause -1

set terminal wxt enhanced font "arial,10" fontscale 1.0 size 600, 600

set border linewidth 1.5 linecolor rgb '#FFA500'  # 橙色
set tics textcolor rgb '#FFA500'  # 橙色
set style line 1 lc rgb '#008000' lt 1 lw 2  # 绿色

# 定义分段函数
z(x,y) = x*y != 0 ? x*y : 1

# 设置绘图范围和密度
set xrange [-30:30]
set yrange [-30:30]
set zrange [-30:30]
set isosamples 80, 80  # 设置绘图密度

# 设置视角
set view 60, 10, 1.0, 1.0

# 绘制 z(x, y)
splot z(x, y) with lines linestyle 1

# 添加过 x=0 点且平行于 y 轴的坐标轴
set arrow from 0, graph 0, first 0 to 0, graph 1, first 0 nohead linecolor rgb '#FFA500'

pause -1

#### 反例二

$$f(x, y)=\left\{\begin{array}{c} \left(x^{2}+y^{2}\right) \sin \frac{1}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0) \end{array}\right.$$

$$f_{x}^{\prime}(0,0)=\lim \limits_{x \rightarrow 0} \frac{f(x, 0)-f(0,0)}{x}=$$

$$\lim \limits_{x \rightarrow 0} \frac{x^{2} \sin \frac{1}{x^{2}}-0}{x}=x \sin \frac{1}{x^{2}}=0$$

$$f_{y}^{\prime}(0,0) = 0$$

$$\lim \limits_{\Delta x \rightarrow 0} \frac{f(\Delta x, \Delta y)-f(0,0)-\left[f_{x}^{\prime}(0,0) \Delta x+f_{y}^{\prime}(0,0) \Delta y\right]}{\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}} =$$

$$\lim \limits_{\substack{\Delta x \rightarrow 0 \\ \Delta y \rightarrow 0}} \frac{f(\Delta x, \Delta y)}{\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}}=$$

$$\lim \limits_{\Delta x \rightarrow 0} \sqrt{(\Delta x)^{2}+(\Delta y)^{2}} \cdot \sin \frac{1}{(\Delta x)^{2}+(\Delta y)^{2}}=0$$

$$f^{\prime \prime}(x, 0) = x^{2} \sin \frac{1}{x^{2}} \Rightarrow$$

$$f_{x}^{\prime}(x, 0)=2 x \cdot \sin \frac{1}{x^{2}}+x^{2} \cos \frac{1}{x^{2}} \cdot \frac{-2 x}{x^{4}}$$

$$f_{x}^{\prime}(x, 0)=2 x \cdot \sin \frac{1}{x^{2}}-\frac{2}{x} \cos \frac{1}{x^{2}} \Rightarrow$$

$$\lim \limits_{x \rightarrow 0} f_{x}^{\prime}(x, 0)=\frac{-2}{x} \cos \frac{1}{x^{2}} \Rightarrow \text{极限不存在}$$

$$\lim \limits_{x \rightarrow 0} f_{x}^{\prime}(x, 0) \neq f_{x}^{\prime}(0,0)$$

$$\lim \limits_{y \rightarrow 0} f_{y}^{\prime}(0, y) \neq f_{y}^{\prime}(0,0)$$