Does partial derivative existence imply continuity in one variable?

[tmlfan_17]

You know that f_x(x₀,y₀) exists. What can you tell about the continuity of g(x)=f(x, y₀) at x=x₀?

I know the answer is that it is continuous but I just wanted somebody to confirm why.

 

[mathman]

Since yo is fixed, you simply have a function of one variable (x), so apply what you know about a function of one variable and its derivative.