= Strictly increasing function
{title2=$x<y\implies f(x)<f(y)$}
= Strictly increasing
{synonym}
A <real-valued function> $f:D\to\mathbb R$ is a <strictly increasing function> if every pair $x<y$ in its <domain of a function> satisfies $f(x)<f(y)$. It is therefore <injective>, and each output has at most one <preimage>. Its <inverse function> on its <image of a function> is also <strictly increasing>. Every <strictly increasing function> is a <nondecreasing function>; the converse fails when outputs repeat.
The <polynomial> $f(x)=x^3$ is a <strictly increasing function> on the <real line>, although its <derivative> $f'(x)=3x^2$ is zero at zero. Indeed, for $x<y$,
$$
y^3-x^3=(y-x)(x^2+xy+y^2)>0,
$$
because $x^2+xy+y^2=(x+y/2)^2+3y^2/4$ can vanish only when $x=y=0$. Thus a positive <derivative> everywhere is sufficient on a <real interval>, but is not necessary for a <strictly increasing function>.
Strict <monotonicity> does not ensure <continuity>: $g(x)=x$ for $x<0$ and $g(x)=x+1$ for $x\geq0$ is a <strictly increasing function> with a <jump discontinuity>. Its <image of a function> omits $[0,1)$, so the <inverse function> exists only on the actual <image of a function>, rather than on all the <real numbers>.
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