Is the curve always continuous if both $x=f(t)$, $y=g(t)$ are continuous?

Question

Initially I thought some functions like ( $x=t^2$, $y=t^4$) simply don't apply to this, but is there like a more general definition for continuity, or should we specify which function we are referring to, so $f$ and $g$ are continues but the curve isn't, and do they have direct correlation ($f/g$ and the curve ) in terms of continuity.

Answer 1

The curve you are considering is given by the function $C$ whose domain is $\mathbb{R}$ and whose codomain is $\mathbb{R}^2$ (we write this as $C : \mathbb{R} \rightarrow \mathbb{R}^2$). It is explicitly given by: $$ C(t) = (t^2, t^4) $$ You can also write this as $C(t) = (x(t), y(t))$ or $C(t) = (f(t), g(t))$, where $x(t) = f(t) = t^2$ and $y(t) = g(t) = t^4$.

What it means for this function $C$ to be continuous at a point $a$, as you pointed out, is that it satisfies the condition: $$ \lim \limits_{t \rightarrow a} C(t) = C(a). $$ Using the more technical definition of limits, you can show that this is equivalent to each of the components being continuous. Specifically, if we say that $C(t) = (f(t), g(t))$, then: $$ \lim \limits_{t \rightarrow a} C(t) = C(a) \qquad \text{is equivalent to} \qquad \lim \limits_{t \rightarrow a} f(t) = f(a) \text{ and } \lim \limits_{t \rightarrow a} g(t) = g(a). $$ So, the answer to your question is: a curve is continuous if and only if its components are all continuous.