Solved Problem on Cauchy-Riemann Equations
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b)   \( \displaystyle w=\text{Re}\;z \)

Writing the function as
\[ w=\text{Re}\;z=\text{Re}(x+iy)=x \]
Condition 1: The function w is continuous everywhere in the complex plane.

The Cauchy-Riemann Equations are given by
\[ \bbox[#99CCFF,10px] {\begin{gather} \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\\[5pt] \frac{\partial u}{\partial y}=-{\frac{\partial v}{\partial x}} \end{gather}} \]
Identifying the functions u(x, y) and v(x, y)
\[ \begin{array}{l} u(x,y)=x\\ v(x,y)=0 \end{array} \]
Calculating the partial derivatives
\[ \begin{array}{l} \dfrac{\partial u}{\partial x}=1\\[5pt] \dfrac{\partial v}{\partial y}=0\\[5pt] \dfrac{\partial u}{\partial y}=0\\[5pt] \dfrac{\partial v}{\partial x}=0 \end{array} \]
Condition 2: The derivatives are continuous everywhere in the complex plane.

Applying the Cauchy-Riemann Equations
\[ \begin{gather} \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\\ 1\neq 0 \end{gather} \]
\[ \begin{gather} \frac{\partial u}{\partial y}=-{\frac{\partial v}{\partial x}}\\ 0=0 \end{gather} \]
Condition 3: The function w does not satisfy the Cauchy-Riemann Equations.

The function w and the derivatives are continuous, but the function does not satisfy the Cauchy-Riemann Equations, the function w is not analytic and is not differentiable in the complex plane. .
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