Date: 9/1/96 at 19:18:56 From: Jim Hayes Subject: Continuity Hello Dr.Math, My teacher friend states that the absolute value function is not continuous. His reasoning is based on a definition he uses in secondary school: "A discontinuous function is a function specified by different rules over different values for its domain." As an example, when looking at the graph of the absolute value function, he says this graph could also be the graph of the union of two functions, f(x)=x, where the domain is restricted to values equal to or greater than zero, and g(x)=-x, where the domain is restricted to values equal to or less than zero. Of course the graphs are identical, but I believe that his definition is all wet. What his definition leads to is that any graph that changes direction in a manner which is not smooth, is discontinuous. What do think of his reasoning? Jim Hayes
Date: 9/2/96 at 12:39:22 From: Doctor Ken Subject: Re: Continuity Hello! You are absolutely correct. The absolute value function is definitely continuous. The "different rules for different parts" criterion might work in some cases (or rather, it can give you clues for which functions you should look at more closely), but it's not the definition of continuity (for more information on continuity, you can browse our archives). Note further that you can define the absolute value function a different way: |x| = Sqrt(x^2). This produces the exact same function, so if one function is continuous then the other one is. Using the "different rules for different parts" criterion would give you different results for these two ways of defining, so something's fishy. A better way of thinking about continuity (although it too is not the definition) is whether you can draw the function without picking up your pencil. The actual definition? Given x, the limit as c approaches x exists and it's equal to f(x) if and only if f is continuous at x. Then you say f is continuous (all over) if it's continuous at every x. Hope this helps. Oh, by the way, what the absolute value function is _not_ is differentiable. That means it's got a sharp edge. -Doctor Ken, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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