1. The problem statement, all variables and given/known data I have always been comfortable with proving continuity of a function on an interval, but I have been running into problems proving that a function is continuous at a point in it's domain. For example: Prove [itex]f(x) = x^2[/itex] is continuous at [itex]x = 7[/itex]. 2. Relevant equations We will be using the delta epsilon definition of continuity here. 3. The attempt at a solution Let [itex]f(x) = x^2[/itex] and [itex]\varepsilon > 0[/itex]. Choose [itex]\delta[/itex]= ________ (usually we choose [itex]\delta[/itex] last, so I am just leaving it blank right now). Now, if [itex]|x - y| = |7 - y| = |y - 7| < \delta[/itex], then [itex]|f(x) - f(y)| = |49 - y^2| = |y^2 - 49| = |y + 7||y - 7|.[/itex] This is where it gets a little awkward for me. I know that I may say [itex]|y - 7| < \delta[/itex], but what do I do with the [itex]|y + 7|[/itex]? Could I say that [itex]|y + 7| < \delta + 14[/itex]? Then I would have to choose a [itex]\delta[/itex] such that [itex]\delta (\delta + 14) = \varepsilon[/itex]. Thank you for your help anyone!