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Differentiability definitions

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  • Continuity
    A property where a graph has no holes, jumps, or asymptotes, allowing it to be drawn without lifting a pencil.
  • Differentiability
    A condition where a function is both continuous and smooth, ensuring the derivative exists at all points.
  • Derivative
    A measure of how a function changes as its input changes, representing the slope of the tangent line at a point.
  • Sharp Corner
    A point on a graph where the direction changes abruptly, causing the tangent line to change suddenly.
  • Piecewise Function
    A function defined by different expressions over different intervals, often requiring special checks at boundaries.
  • Limit
    A value that a function approaches as the input approaches a certain point from either side.
  • Polynomial
    An algebraic expression with terms consisting of variables raised to whole number powers, always continuous and differentiable.
  • Boundary Point
    A value where two pieces of a piecewise function meet, often requiring checks for continuity and differentiability.
  • Tangent Line
    A straight line that touches a curve at a single point, matching the curve's slope at that point.
  • Jump
    A discontinuity where the graph suddenly moves from one value to another, breaking continuity.
  • Asymptote
    A line that a graph approaches but never touches, indicating a type of discontinuity.
  • Smoothness
    A quality of a graph where there are no abrupt changes in direction, ensuring differentiability.
  • Left-Hand Derivative
    The slope of the tangent line as the input approaches a point from the left side.
  • Right-Hand Derivative
    The slope of the tangent line as the input approaches a point from the right side.
  • Value of Interest
    A specific input where continuity and differentiability are checked, often at boundaries in piecewise functions.