Piecewise Function Graph with Open Ends

ROLE: Act as a maths teacher who graphs piecewise functions with perfectly correct endpoint dots.

CONTEXT: I teach [CLASS/GRADE] under [BOARD or EXAM e.g. CBSE / ICSE / JEE]. Students must see exactly where each piece starts and stops and whether the endpoint is included.

YOUR TASK:
1. Plot each piece of the function only over its own interval, on labelled x and y axes with a light grid.
2. At every breakpoint, use a FILLED dot where the value is included and an OPEN dot where it is excluded.
3. Label each piece with its rule and its interval near that segment.
4. Show the breakpoint x-values on the axis.
5. Make it clear whether the function is continuous or has a jump at each breakpoint.

INPUT: Here is the piecewise function: [PASTE IT HERE, e.g. f(x) = x + 1 for x < 2; f(x) = 5 for x >= 2].

HOW TO WORK IT OUT: For each breakpoint, compute the value each adjoining piece approaches and decide, from the inequality signs (< vs <=, > vs >=), which side gets the filled dot and which gets the open dot. Show this reasoning. VERIFY two ways: confirm each piece is drawn only on its stated interval, and confirm at each breakpoint exactly one filled dot exists when the function is defined there (and note any jump). Never put two filled dots at the same x; never extend a piece past its interval.

OUTPUT FORMAT:
1. ENDPOINT ANALYSIS: for each breakpoint, the left value, right value, and which dot is open/filled.
2. CONTINUITY NOTE: continuous or jump at each breakpoint.
3. THE GRAPH: one clean labelled image with each piece, correct dots, rules and intervals shown.
4. CHECK: one line confirming no piece overruns its interval.

RULES: Filled versus open dots must match the inequalities exactly. Each piece confined to its interval. Axes labelled, breakpoints marked. Plain white background.

SIGNATURE: Sign the bottom-right of every page with an elegant handwritten signature "[YOUR NAME]" with a small underline flourish.