Define and plot piecewise functions with domain intervals. Shows open/closed endpoints, continuity checks, and limits.
A piecewise function is one formula on part of its domain and a different formula on the rest, and all of its interesting behavior sits at the seams between the parts. This tool takes each branch as a formula plus a domain interval, draws it only over that interval, marks boundary points with filled or hollow circles, and computes the one-sided limits at every breakpoint to report whether the function is continuous there. The point people most often miss: the value at a breakpoint belongs to whichever branch includes it, so a jump, a hole, or an isolated point can all hide at the same x-value.
A concrete input and expected output from the current implementation.
Input
f(x) = x^2 for x < 1; 3 - x for x >= 1
Expected output
Two branches plotted: the parabola y = x^2 drawn only left of x = 1, ending in a hollow circle at (1, 1); the line y = 3 - x drawn from x = 1 onward, starting with a filled circle at (1, 2). Limits report: left limit 1, right limit 2, value 2, so the limit at x = 1 does not exist and the function is discontinuous there (a jump from height 1 to height 2).
As x approaches 1 from below, x^2 approaches 1, while the branch that owns x = 1 gives 3 - 1 = 2. Because the two sides approach different heights, the graph jumps and continuity fails at x = 1.
Each row pairs an expression with an x-interval, and the graph evaluates that expression only at x-values inside the interval. A closed boundary (<= or >=) plots a filled circle at the branch's value; an open boundary (< or >) plots a hollow circle at the value the branch approaches; where no interval applies, nothing is drawn, so the function visibly ends at its outermost bounds.
At every x where the interval list changes, the tool forms the one-sided limits from the branches on either side and compares them with each other and with the function's defined value there. Matching limits with a matching value mean the function is continuous; matching limits with a missing or different value mean a hole (removable discontinuity); unequal limits mean a jump. The verdict for each breakpoint is reported alongside the graph.
Because neither branch owns the breakpoint. With strict inequalities on both sides the function has no value at x equal to the boundary, so even though the left and right limits agree, continuity fails at an undefined point. Change one inequality to <= or >=, or add an explicit single-point branch, and the gap closes.
No. Each branch is confined to its own interval, and x-values covered by no interval are outside the function's domain, so nothing is drawn there. If the curve stops abruptly at an edge, that is correct behavior; extend that interval or add another branch rather than expecting the tool to extrapolate.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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