Intermediate value property with no continuity Definition: A real function f has the intermediate value property on an interval I containing [a,b] if f(a) &lt; v &lt; f(b) or f(b) &lt; v &lt; f(a); that is, if v is between f(a) and f(b), there is between a and b a c ∈ [a,b] such that f(c) = v.

What is an example function that has this property, and how do I prove that a strictly increasing function f:[a,b]→ R which has this property is continuous on [a,b]?
Presumably you wanted an example of a discontinuous function satisfying IVP. If $f$ is differentiable then the Mean Value Theorem shows that $f'$ has IVP, although $f'$ need not be continuous.
An example is the identity.

Hint (for $f$ increasing): $f$ must be surjective in $[a,b]$ for all $[a,b]\subset I$