Probe continuity at any point with a live three-condition checklist — f(c) defined, two-sided limit exists, f(c) equals the limit. Slide c through holes, jumps, asymptotes, and staircases and watch each row flip pass or fail in real time.
Watch left and right limits race toward each other as epsilon shrinks. Switch between seven canonical cases — continuous, hole, jump, infinite asymptote, oscillating, and one-sided — and read the verdict for both the limit and the continuity at c.
Move x0 along the graph and watch three pictures of one number lock together — the slope of the tangent on f, the height of f-prime at x0, and the numeric derivative value. Snap directly to roots, extrema, and inflection points to see why each one matters.
Draw a secant between any two points on a smooth curve and the tool finds every interior c where the tangent has matching slope — the parallel-tangent guarantee at the heart of the Mean Value Theorem. Six function families illustrate the single-c, multi-c, and exact-midpoint cases.
Find every critical point of a smooth function on a chosen interval and watch the second-derivative test classify each as local max, local min, or inflection. Six families show single-extremum cases, W-shapes, and touch zeros where the test falls back to the first-derivative test.
See both halves of the Fundamental Theorem of Calculus on one graph — slide x to grow the shaded area under f and watch the accumulator F match the area exactly, while the slope of F at every x equals f at the same point.
Approximate a definite integral with rectangles or trapezoids, then watch the error shrink as you increase the partition count. Switch between left, right, midpoint, and trapezoid rules to see why some converge as one over n and others as one over n squared.