Plotting a piecewise defined function

Define $f(x)$ by

1. Sketch the graph of $f$ for $c=1$.
2. State precisely why $f$ is discontinuous at $x=1$ when $c=1$.
3. For what value of $c$ is $f$ continuous for all real numbers?
4. Sketch the graph of $f$ when for your choice of $c$ from part 3.

Note: I think this problem should be very doable by hand. It might be fun to produce a Desmos plot to answer the question, though.

2. The graph is discontinuous because the limit from the left and right do not converge on the same number.

3. When $c=6$ the graph is continuous for all real numbers.