Continuity and Differentiability Cheat Sheet

Key Definitions

• Continuity: A function f(x) is continuous at x = a if:

• f(a) is defined

• lim (x→a) f(x) exists

• lim (x→a) f(x) = f(a)

• Differentiability: A function f(x) is differentiable at x = a if:

• The derivative f'(a) exists.

Important Concepts

Continuity

• Types of discontinuities:

• Removable: Limit exists but not equal to function value.

• Jump: Limit does not exist due to different left/right limits.

• Infinite: Function approaches infinity.

Differentiability

• A function must be continuous to be differentiable.

• Sharp corners or vertical tangents indicate non-differentiability.

Key Formulas

• f'(x): Derivative of f at x

• h: Small increment approaching zero

Common Mistakes to Avoid

Common Mistake: Assuming a function is differentiable at a point without checking continuity.

Memory Tricks

Tip: "C-D-C" — Continuous means Defined, Converges (limit exists).

Key Diagrams

Continuity:
  f(a)  ----o
          |
          |     lim
          |------>
          |
          |     lim
          |------>
          |
          a
Differentiability:
  |\
  | \
  |  \
  |   \
  |    \
  |     \
  |      \
  |_______\________