Recall that a function [math]f[/math] is [i]continuous [/i]at a point [math]c[/math] if for every [math]\epsilon>0[/math] there exists a [math]\delta>0[/math] such that if [math]\left|x-c\right|<\delta[/math], then [math]\left|f\left(x\right)-f\left(c\right)\right|<\epsilon[/math].[br][br]Further, we say that [math]f[/math] is [i]uniformly [/i]continuous if for every [math]\epsilon>0[/math] there exists a [math]\delta>0[/math] such that if [math]\left|x-y\right|<\delta[/math], then [math]\left|f\left(x\right)-f\left(y\right)\right|<\epsilon[/math].[br][br]This visualization allows one to fix an [math]\epsilon>0[/math] and adjust [math]\delta>0[/math] so that the continuity conditional is satisfied. In addition, one can move the point [math]c[/math] on the [math]x[/math]-axis to see the effect on [math]\delta[/math] and explore the idea of uniform continuity.

Using the applet work on the following questions:[br][br][list=1][*]Starting with [math]f\left(x\right)=x^2[/math] position the point [math]c=0.3[/math] and let [math]\epsilon=0.05[/math], what is the largest value of [math]\delta[/math] that will satisfy the continuity conditional?[/*][*]What is the effect of moving [math]c[/math] to the right?[/*][*]Change the function to [math]f\left(x\right)=2x[/math]. Position the point [math]c=0.2[/math] and let [math]\epsilon=0.05[/math], what is the largest value of [math]\delta[/math] that will satisfy the continuity conditional? What is the effect of moving the point [math]c[/math] on [math]\delta[/math]?[/*][*]What is it about these two functions that causes the different effect on [math]\delta[/math]?[/*][*]Returning to the function [math]f\left(x\right)=x^2[/math], for a particular choice of [math]\epsilon[/math] can we ever find a [math]\delta[/math] that will work for all [math]c[/math]? What about the function [math]f\left(x\right)=2x[/math]?[/*][*]Again, returning to [math]f\left(x\right)=x^2[/math], if we restrict [math]c[/math] to the interval [math]\left[0,\frac{1}{2}\right][/math] can we find a single [math]\delta[/math] that will work for all [math]c[/math]?[/*][*]Consider the function [math]f\left(x\right)=\sin\left(x\right)[/math]. For a given [math]\epsilon[/math] can we find a single [math]\delta[/math] that works for all [math]c[/math]? What about for [math]f\left(x\right)=\sin\left(x^2\right)[/math]? What's the difference?[br][/*][/list]