Limits and continuity
Continuity does not imply differentiability. \(\abs x\) is continuous and not differentiable at \(0\), and Weierstrass’s function is continuous on \(\RR\) and differentiable nowhere.
Differentiability does not imply continuous differentiability. \(f(x)=x^2\sin(1/x)\) for \(x\neq0\), \(f(0)=0\), is differentiable on \(\RR\), and \(f'\) is not continuous at \(0\).
Continuity on a bounded interval does not imply uniform continuity. \(1/x\) is continuous on \((0,1)\) and not uniformly continuous; a continuous function on a closed bounded interval is uniformly continuous.
A continuous function on a bounded interval need not attain its bounds. \(x\) on \((0,1)\) attains neither its supremum nor its infimum; a continuous function on a compact set attains both.
Sequential criterion for continuity. If \(\lim_n f(x_n)\) exists for every sequence \(x_n\to a\), then \(f\) is continuous at \(a\): interleaving two sequences shows that all these limits are equal, and the constant sequence \(x_n=a\) shows that the common limit is \(f(a)\). Hence two sequences \(x_n\to a\) and \(y_n\to a\) with \(\lim f(x_n)\neq\lim f(y_n)\) show that \(f\) is not continuous at \(a\).
Sequences and series
Terms tending to zero do not imply convergence. \(\sum 1/n\) diverges.
Convergence does not imply absolute convergence. \(\sum (-1)^{n+1}/n\) converges and \(\sum 1/n\) diverges; by the Riemann rearrangement theorem, the terms of a conditionally convergent series can be rearranged to converge to any real number.
A convergent sequence need not be eventually monotone. \((-1)^n/n\) converges to \(0\) and is not eventually monotone.
Pointwise convergence does not preserve continuity. \(x^n\) on \([0,1]\) converges pointwise to the function equal to \(0\) on \([0,1)\) and \(1\) at \(1\).
Uniform convergence does not preserve differentiability. \(f_n(x)=\sqrt{x^2+1/n}\) is differentiable and converges uniformly on \(\RR\) to \(\abs x\). Uniform convergence preserves continuity and Riemann integrability; if \(f_n\to f\) pointwise, each \(f_n\) is continuously differentiable, and \(f_n'\) converges uniformly, then \(f\) is differentiable and \(f'=\lim f_n'\).
Derivatives and integrals
A derivative need not be continuous. The derivative of \(x^2\sin(1/x)\), extended by \(0\), is discontinuous at \(0\). Every derivative has the intermediate value property (Darboux’s theorem), so a function without it, such as a step function, is not a derivative.
A bounded function need not be Riemann integrable. The indicator function of \(\QQ\intersect[0,1]\) is bounded and not Riemann integrable.
Lebesgue’s criterion. A function on \([a,b]\) is Riemann integrable if and only if it is bounded and its set of discontinuities has measure zero; in particular its points of continuity are dense in \([a,b]\).
\(f' = 0\) implies \(f\) constant only on an interval. On \((0,1)\union(2,3)\), the function equal to \(0\) on \((0,1)\) and \(1\) on \((2,3)\) has derivative \(0\) and is not constant.
Algebra and linear algebra
Repeated eigenvalues do not prevent diagonalizability. The identity matrix \(I_2\) has the repeated eigenvalue \(1\) and is diagonal.
A real square matrix need not have a real eigenvector. The rotation \(\matt{0}{-1}{1}{0}\) has characteristic polynomial \(x^2+1\) and no real eigenvalue.
Equal characteristic polynomials do not imply similarity. \(\matt{0}{0}{0}{0}\) and \(\matt{0}{1}{0}{0}\) both have characteristic polynomial \(x^2\) and are not similar; the \(4\times4\) matrices \(J_2(0)\oplus J_2(0)\) and \(J_2(0)\oplus J_1(0)\oplus J_1(0)\) also have the same minimal polynomial \(x^2\) and are not similar.
A group of prime power order need not be abelian. Groups of order \(p\) and \(p^2\) are abelian, and the dihedral group \(D_4\) of order \(8\) is not.