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Consider the function \(f\) defined on \(\mathbb{R}\) by:$$ f(x) = \begin{cases} 1 & \text{if } x \ge 0 \\ 0 & \text{if } x < 0 \end{cases} $$The graph of \(f\) is shown below:
  1. Graphically, conjecture whether the function \(f\) is continuous at \(x=0\).
  2. Prove your conjecture using the mathematical definition of continuity.

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