Does the Cauchy problem $x'=\operatorname{sgn}(t), x(0)=0$ have a solution?

Question

Check if Cauchy's Problem has a solution: $$ x'=\operatorname{sgn}(t), x(0)=0 $$

My solution: by definition of the $\operatorname{sgn}$ fuction $$ x'=1 \Rightarrow x(t)=t+C_1 $$ $$ x'=-1 \Rightarrow x(t) = -t + C_2 $$ The solution doesn't exist because of lack of continuity.

However I'm not sure if it's right solution.