There are many version of criterion for a function to be holomoprhic when CR equation holds ,for example in Stein and Shakarchi's complex analysis book has a version that :
Theorem $2.4$ Suppose $f=u+i v$ is a complex-valued function defined on an open set $\Omega .$ If $u$ and $v$ are continuously differentiable and satisfy the Cauchy-Riemann equations on $\Omega$, then $f$ is holomorphic on $\Omega$ and $f^{\prime}(z)=\partial f / \partial z$
I paste the proof below:
Proof. Write $$ u\left(x+h_{1}, y+h_{2}\right)-u(x, y)=\frac{\partial u}{\partial x} h_{1}+\frac{\partial u}{\partial y} h_{2}+|h| \psi_{1}(h) $$ and $$ v\left(x+h_{1}, y+h_{2}\right)-v(x, y)=\frac{\partial v}{\partial x} h_{1}+\frac{\partial v}{\partial y} h_{2}+|h| \psi_{2}(h) $$ where $\psi_{j}(h) \rightarrow 0$ (for $j=1,2$ ) as $|h|$ tends to 0 (in this step we use differentiable of $u,v$), and $h=h_{1}+i h_{2}$. Using the Cauchy-Riemann equations we find that $$ f(z+h)-f(z)=\left(\frac{\partial u}{\partial x}-i \frac{\partial u}{\partial y}\right)\left(h_{1}+i h_{2}\right)+|h| \psi(h) $$ where $\psi(h)=\psi_{1}(h)+\psi_{2}(h) \rightarrow 0$, as $|h| \rightarrow 0$. Therefore $f$ is holomorphic and $$ f^{\prime}(z)=2 \frac{\partial u}{\partial z}=\frac{\partial f}{\partial z} $$
There are some observation here,First we can find that to check $f$ holomorphic at some point,we only need to check that:
(1)$f$ has $u,v $ is $C^1$ at that point
(2) CR equation holds at that point.
I found in the proof,we only use the fact that $u,v$ is differentiable ,do not need $u,v$ be continuous differentialbe at that point correct?
