continuity of function defined by composition of functions from $\mathbb{R^2} \rightarrow \mathbb{R^2}, \mathbb{R} \rightarrow \mathbb{R}$

Question

The question: let $g, h$ be continuous functions on $\mathbb{R}$ and f continuous on $\mathbb{R^2}$. For each t, define $y(t) = f(g(t), h(t))$. Prove that $y$ is a continuous function on $\mathbb{R}$.

My ideas: use sequential continuity....? I don't think just stating "the composition of continuous functions is also a continuous function" is going to be enough!

Answer 1

The product of two functions is continuus, so $g\times h:\mathbb{R}^2\rightarrow \mathbb{R}^2$ defined by $(g\times h)(x,y)=(g(x),h(y))$ is continuous. $i:\mathbb{R}\rightarrow \mathbb{R}^2$ defined by $i(t)=(t,t)$ is continuous. The composition of two continuous function is continuous so $y=f\circ (g\times h)\circ i$ is continuous.