How do you prove that the function f ( x ) = x - 5 | x - 5 | is continuous everywhere but x=5?

Question

How do you prove that the function             f              (        x        )              =                  x        -        5                    |        x        -        5        |             is continuous everywhere but x=5?

trollbabyfeetwa · Accepted Answer

Use the definition of             |      u      |        =          {                                    -            u                                              if                                            u            &amp;lt;            0                                                u                                              if                                            u            ≥            0                               to rewrite f as a piecewise function.Explanation:            f              (        x        )              =          {                                    -            1                                              if                                            x            &amp;lt;            5                                                1                                              if                                            x            &amp;gt;            5                              So, f is constant on             (      -      ∞      ,      5      )      , hence continuous on             (      -      ∞      ,      5      )      and f is constant on             (      5      ,      ∞      )      , hence continuous on             (      5      ,      ∞      )      f(5) is not defined, so f is not continuous at 5.

How do you prove that the function f(x)=(x-5)/ |x-5| is continuous everywhere but x=5?

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