For the function $f(z)=1$ $(z\in \mathbb{C})$ and C is an arbitrary contour from any fixed point ${z}_{1}$ to any fixed point ${z}_{2}$ in the $z$ plane. Use parametric representations for $C$ to evaluate $\int _{C}^{}f(z)dz$.
Here is my solution: Let ${z}_{1}$ and ${z}_{2}$ are two fixed point and $C$ is straight line. Then $z=(1-t){z}_{1}+t{z}_{2}$ where $t\in \left[0,1\right]$, from here $dz=-{z}_{1}dt+{z}_{2}dt$. Then we get
$\int _{{z}_{1}}^{{z}_{2}}f(z)dz=\int _{0}^{1}1({z}_{2}dt-{z}_{1}dt)=\int _{0}^{1}{z}_{2}dt-\int _{0}^{1}{z}_{1}dt={z}_{2}-{z}_{1}$
Hence,
$\int _{C}^{}f(z)dz=\int _{{z}_{1}}^{{z}_{2}}f(z)dz=\int _{{z}_{1}}^{{z}_{2}}1dz={z}_{2}-{z}_{1}$
Is that my solution right ? I feel that I'm missing something. Any help will be appreciated.
Your proof is fine but restricted to the case where $C$ is a staight line. You are asked to prove the result for an arbitrary contour.
According to the definition given in the comments, a contour $C$ is the concatenation of smooth curves $C_1,\dots,C_K$. The curve $C_k$ has endpoints $c_{k-1},c_{k}$ and is parametrized by the $C^1$ function $\gamma_i:[0,1] \mapsto \mathbb{C}$. In particular, $c_0 = z_1$ and $c_K = z_2$.
By the formula for countour integration, for each $C_k$:
$$\int_{C_k} f(z) dz = \int_0^1 f(\gamma_k(t)) \gamma_k'(t) dt.$$
For the specific case $f(z) =1$, it yields:
$$\int_{C_k} f(z) dz =\int_0^1 \gamma_k'(t) dt=\left[ \gamma_k (t)\right]_0^1=c_k-c_{k-1}.$$
The concatenation of the smooth curves $C_k$ to create $C$ leads to:
$$\int_C f(z) dz = \sum_{k=1}^K \int_{C_k}f(z) dz= \sum_{k=1}^Kc_k-c_{k-1}$$
This sum is telescopic, and we are left with, as wanted:
$$\int_C f(z) dz = c_K-c_0=z_2-z_1.$$
Further note : this shows that the countour integration of the constant function $1$ is path-independent, it only depends on the endpoints. This property is in fact a characterization of holomorphic functions, as shown here. Holomorphic function is the complex equivalent of being derivable for real ones, so it can be seen as an extension of the fundamental theorem of calculus to the complex domain.