Answer
$f(x)$ has a continuous extension at $x=1$, whose value is $1.3334$.
Work Step by Step
$$f(x)=\frac{x-1}{x-\sqrt[4]x}$$
The graph is enclosed below.
Condition to have a continuous extension for $f(x)$ at $x=c$: $\lim_{x\to c}f(x)$ must exist.
Here, $\lim_{x\to1}f(x)$ does seem to exist, because as $x$ approaches $1$ from the left and the right, $f$ approaches a value of approximately $1.3334$. That means $\lim_{x\to1}f(x)=1.3334$.
So function $f$ can be extended to include the value $f(1)=1.3334$ so that $\lim_{x\to1}f(x)=f(1)$ and $f$ would then be continuous at $x=1$.
