The Intermediate Value Theorem.[br]If [math]f[/math] is a continuous function on the interval [math][a,b][/math], and if [math]y_0[/math] is any value between [math]f(a)[/math] and [math]f(b)[/math], then [math]y_0 = f(c)[/math] for some [math]c[/math] in [math][a,b][/math][br][br]This applet has several functions to use for exploring the intermediate value theorem. The "next" button changes the function. You can move the points [math]a[/math] and [math]b[/math]. A slider then moves [math]y_0[/math] between [math]f(a)[/math] and [math]f(b)[/math]. If a [math]c[/math] value can be found with [math]f(c)=y_0[/math] it is shown.

Explore the various functions with [math]a[/math] and [math]b[/math] set where [math]f(x)[/math] is continuous or not continuous.[br][br]Are [math]c[/math] values found if [math]f(x)[/math] is not continuous?[br][br]Are there functions and boundaries where multiple [math]c[/math] values could work for [math]f(c) = y_0[/math] ?[br][br]Note the statement when no [math]c[/math] value is found. Is this logically correct?