= Continuous image of a connected space
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= Continuous images of connected spaces
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If $f:X\to Y$ is a <continuous map> and $X$ is a <connected space>, then its <image> $f(X)$, with the <subspace topology> from $Y$, is a <connected subset> of $Y$. The whole <codomain> $Y$ need not be <connected>; the assertion concerns the values actually attained. No <injectivity> is required.<General Topology by Tom Leinster>[p. 70: a continuous image is connected]{ref}
For example, the <continuous function> $f(x)=x^2$ maps the <connected> <closed interval> $[-2,2]$ onto the <connected> <closed interval> $[0,4]$, even though opposite inputs have the same output. The same <continuous function> maps the disconnected <union> $[-2,-1]\cup[1,2]$ onto $[1,4]$. Thus a <connected> <image> does not imply a <connected> source: a <continuous map> can merge different <connected components>.
For a <continuous function> $r:[0,1]\to\mathbb R$ representing a continuous reading, $r(0)=0$ and $r(1)=1$ force some input with $r(t)=1/2$. Its <image> is a <connected subset of the real line>, hence a <real interval> containing both endpoint readings. For the synthetic model $r(t)=t^2$, the missing-value test is explicit: $r(1/\sqrt2)=1/2$. A step reading equal to $0$ for $t<1/2$ and $1$ for $t\ge1/2$ skips $1/2$, but fails <continuity> at $t=1/2$.
\Image[/continuous-readings-intermediate-values.png]
{title=A <continuous function> attains intermediate readings; a step that skips them violates <continuity>}
{id=continuous-readings-intermediate-values}
{description=Synthetic mathematical models, not measured data. The <continuous function> $r(t)=t^2$ attains $1/2$ at $t=1/\sqrt2$; the step attains only $0$ and $1$.}
Proof outline: a <separation of a topological space> of $f(X)$ into nonempty <open subsets> pulls back under $f$ to nonempty <open subsets> of $X$. Their <intersection> is empty and their <union> is $X$, contradicting <connectedness>. This also shows why a <continuous map> sends each <connected component> of its source into a single <connected component> of its target.
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