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- We can define an interval to be a
*The image of an interval under a continuous map is also an interval.***Proof**

- If
*f*(*a*),*f*(*b*) ∈*f*(*I*) and*y*lies between them, then by the*intermediate value theorem*there is an*x*between*a*and*b*with*f*(*x*) =*y*.

- Recall that intervals come in various types.
*Open*: like (*a*,*b*), (*a*, ∞), (-∞, ∞), ...*Closed*: like [*a*,*b*]*Half-open*: like [*a*,*b*), (*a*,*b*], [*a*, ∞), ...**Theorem**

*The image of a closed, bounded interval under a continuous map is closed and bounded.***Proof**

- By the theorem of the previous section, the image of an interval
*I*= [*a*,*b*] is bounded and is a subset of [*m*,*M*] (say) where*m*,*M*are the lub and glb of the image. Since the function attains its bounds,*m*,*M*∈*f*(*I*) and so the image is [*m*,*M*].

**Remark**- The images of
*any*of the other intervals can be*any*kind of interval.

**Theorem**

- Any open interval can be mapped to any other open interval (in fact by a continuous
*bijection*):For example, (-π/

_{2}, π/_{2})→ (-∞, ∞) by the*tangent*function.(-∞, ∞)→ (0, ∞) by the

*exponential*function.Any finite open interval can be mapped to any other finite open interval by a suitable

*linear*function. (Exercise) - Any half-open interval can be mapped to any other half-open interval using the same maps as in 1.
- Any open or half-open interval can be mapped to a closed interval.
For example, (-∞, ∞) or [0, ∞)→ [-1, 1] by the

*sine*function.Any open interval can be mapped to a half-open interval.

For example, (-∞, ∞)→ [0, ∞) by the*x*^{2}function.

- Finally, (the tricky one) any half open interval can be mapped onto an open interval.

For example, [0, ∞)→ (-∞, ∞) by the map*x*→*x*sin*x*.

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