Examples counterexamples

• Examples of Continuous Strictly Increasing Functions on ℝ
  The requested property is missing from the title; arctan is continuous and strictly increasing on ℝ, yet bounded and not onto ℝ.
• Examples of Additive-Only and Multiplicative-Only Functions
  See examples of functions that preserve sums but not products, or products but not sums, and learn when an additive example must be pathological.
• Does $f(0)=0$ Imply That $f'(x)=0$?
  Knowing that f(0)=0 tells you only that the graph passes through the origin; it does not determine whether f' exists or what its value is.
• What Is a Counterexample to ∀x∃y (x = 1/y)?
  Over the real numbers, x = 0 is a counterexample: no nonzero real y has reciprocal zero. The answer depends on the domains.
• Examples of 3×3 Matrices With Common Properties
  Browse verified 3×3 matrix examples for symmetry, rank, invertibility, orthogonality, idempotence, nilpotence, and noncommutativity.
• A Simple Example of a Lindelöf Space
  The real line is a simple Lindelöf space: its countable rational-interval basis lets you extract a countable subcover from any open cover.