Answer
No, see explanations.

Work Step by Step
Due to the gap at $a=0$, $\lim_{x\to0^−}k(x)≠\lim_{x\to0^+}k(x)$, we see that the limit $\lim_{x\to0}k(x)$ does not exist. Thus the discontinuity at $a=0$ can not be extended.
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