= Solution
{c}
Use the <Heaviside step function> with the value at the jump chosen as follows:
$$
\boxed{f(x)=\begin{cases}0,&x<0,\\1,&x\geq0.\end{cases}}
$$
For any <open set> in the target <Sorgenfrey line>, its <preimage> is one of $\varnothing$, $\mathbb R$, $(-\infty,0)$, and $[0,\infty)$. All four are <open> in the source <Sorgenfrey line>, so $f$ is <continuous>. In the <Euclidean topology>, the <preimage> of $(1/2,3/2)$ is $[0,\infty)$, which is not <open>. Hence the same <function> is not <continuous> as a usual real-to-real <map>.
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