1. **State the problem:** We are given a piecewise function $$T(x)$$ defined as: $$ T(x) = \begin{cases} 0 & \text{if } x < 0 \\ x & \text{if } 0 \leq x < 1 \\ 1 & \text{if } x \geq 1 \end{cases} $$ We want to understand and analyze this function, including its graph and behavior. 2. **Explain the function segments:** - For $$x < 0$$, $$T(x) = 0$$, which is a constant horizontal line at $$y=0$$. - For $$0 \leq x < 1$$, $$T(x) = x$$, which is a line with slope 1 starting at the origin and going up to but not including $$x=1$$. - For $$x \geq 1$$, $$T(x) = 1$$, which is a constant horizontal line at $$y=1$$. 3. **Important points and continuity:** - At $$x=0$$, $$T(0) = 0$$, so the function is continuous there. - At $$x=1$$, the function value is $$T(1) = 1$$, matching the right segment. 4. **Graph features:** - The graph has a flat segment at $$y=0$$ for $$x<0$$. - A diagonal line from $$(0,0)$$ to $$(1,1)$$. - A flat segment at $$y=1$$ for $$x \geq 1$$. - Hollow circles at $$(0,0)$$ and $$(1,1)$$ indicate the function values are included at those points. 5. **Summary:** This piecewise function smoothly transitions from 0 to 1 as $$x$$ moves from negative values through 0 to values greater than or equal to 1. **Final answer:** The function $$T(x)$$ is defined and continuous as described above with the given piecewise segments and graph behavior.