Answer
The limit is
$$\lim_{(x,y)\to(0,0)}(x+4y+1)=1.$$
The function is continuous on its' domain.
Work Step by Step
This limit is calculated by a simple substitution
$$\lim_{(x,y)\to(0,0)}(x+4y+1)=0+4\times0+1=1.$$
This function is continuous on its domain because for every ordered pair $(x,y)$ we have
$$\lim_{(x,y)\to(x_0,y_0)}(x+4y+1)=x_0+4y_0+1.$$