Prove that $f:\mathbb{R}^n\to\mathbb{R}$ is continuous

Question

Let $f:\mathbb{R}^n\to\mathbb{R}$ such that for every continuous curve, $\gamma:[0,1]\to\mathbb{R}^n$: $f\circ\gamma$ is continuous. Prove that $f$ is continuous.

So I know we shall prove it by a contradiction. Let's assume that $f$ isn't continuous at $x_0$. Then, there's a sequence, $\{x_k\}$ converging to $x_0$ such that $\lim_{k\to\infty} f(x_k) \ne f(x_0)$.

Now, I need to have some curve in order to get a contradiction.

I'd be glad to get help with that.

Thanks.

Answer 1

Construct a curve "joining the points" $x_1$, $x_2$,... with $\gamma(0) = x_1$, $\gamma(1/2) = x_2$,...

Answer 2

Note that you need a sequence $\{x_k\}$ of points converging to $x_0$.

Consider a piecewise-linear curve $\gamma : [0, 1] \to \Bbb R^n$ passing through $x_0$, joining $x_k$'s by straightlines.