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How to Determine the Type of Discontinuous Points?

A function is continuous at one point \(x=a\) means \(\lim_{x\to a}f(x)=f(a)\). This equation has three meanings: if \(f(x)\) is continuous at point \(x=a\),

  • \(f(a)\) is defined
  • \(\lim_{x\to a}f(x)\) exits
  • \(\lim_{x\to a}f(x)=f(a)\)

So any one of above conditions is not satisfied, the function is discontinue at that point. According to which condition is not satisfied, we can classify the type of discontinuous points.

Since the limit exist at a point is equivalent to the left limit equals right limit, i.e \(\lim_{x\to a^-}f(x)= \lim_{x\to a+}f(x) \), the discontinuity conditions (hence discontinuous points) are classified as

  • \(\lim_{x\to a}f(x)\) exits, \(\lim_{x\to a}f(x)\ne f(a)\) \(\Longrightarrow\) removable discontinuous points;
  • \(\lim_{x\to a}f(x)\) exits, \(f(a)\) is not defined \(\Longrightarrow\) removable discontinuous points;
  • \(\lim_{x\to a^-}f(x) \ne \lim_{x\to a+}f(x) \) but both are finite \(\Longrightarrow\) jump discontinuous points.
  • \(\lim_{x\to a^-}f(x)=\pm\infty\) or \( \lim_{x\to a+}f(x)=\pm\infty\) (or both) \(\Longrightarrow\) infinity discontinuous points;
  • \(\lim_{x\to a^-}f(x)\) or \( \lim_{x\to a+}f(x)\) does not exits nor infinity \(\Longrightarrow\) oscillation discontinuous points.

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