1. **Problem:** Given the graph of a function $f$, find the value of $k$ where $f$ is defined at $k$ but $\lim_{x \to k} f(x)$ does not exist. 2. **Understanding the problem:** The limit $\lim_{x \to k} f(x)$ does not exist if the left-hand limit and right-hand limit at $k$ are not equal or if the function has a jump or oscillation at $k$. 3. **From the graph description:** There is a jump/discontinuity at $x = c$ and $f$ is defined at $c$ (since the problem states $f$ is defined at $k$). 4. **Conclusion:** Therefore, $k = c$. **Final answer:** $\boxed{c}$