What does it mean for a function to be discontinuous at a specific point, and why does that happen at point ‘a’?

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Understanding Discontinuity in Functions

Discontinuity in a function arises when the function fails to meet certain criteria at a specific point. For a function to be continuous at a point ‘a’, it must satisfy three main conditions:

  • The function is defined at ‘a’.
  • The limit of the function as it approaches ‘a’ must exist.
  • The limit must equal the function value at ‘a’.

Reasons for Discontinuity

There are several reasons a function may be discontinuous at point ‘a’:

  1. Point Discontinuity: The function is not defined at ‘a’. For instance, if we have a function f(x) = 1/(x-a), it is undefined at x = a, leading to a gap in the graph.
  2. Jump Discontinuity: The left-hand limit and right-hand limit at ‘a’ do not equal each other. An example is the step function, which jumps from one value to another at a specific point.
  3. Infinite Discontinuity: The function approaches infinity as it nears ‘a’. For example, the function f(x) = tan(x) has discontinuities where x = (2n + 1) * π/2, leading the function to blow up to infinity.
  4. Oscillating Discontinuity: The limits oscillate as they approach ‘a’. An example includes functions that exhibit erratic behavior, such as f(x) = sin(1/x) as x approaches zero.

Conclusion

In summary, when analyzing whether a function is discontinuous at a point ‘a’, it’s essential to check the three continuity criteria. If any of these conditions fail, the function is discontinuous at that point. Understanding the type of discontinuity helps us analyze and visualize the behavior of functions in different contexts.

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