11.12
Continuity of a Function
A function is continuous at a point a if three conditions are met: the function is defined at a, the limit of the function as x approaches a exists, a…
Consider a path along which a car moves. If the path is described by the curve of a function, the car’s position at any moment can be understood using the concept of a limit as the x-values approach a point on the curve.
As the car travels along the curve, smooth movement—meaning no breaks or jumps in path—suggests the function is continuous.
If two points approach the same point on the curve from opposite directions and meet smoothly, this indicates that the function is continuous at that point.
A curve can have a discontinuity at a point where the function is undefined, such as when division by zero occurs. As x gets closer to 1, the outputs approach 2. Since this limit exists, the discontinuity is removable, and the function can be redefined at that point to make it continuous.
If the left-hand and right-hand limits at a point don’t match, the function has a discontinuity at that point.
A curve that shoots upward or downward infinitely without touching a vertical line shows a vertical asymptote.
Sudden jumps in a piecewise function mark points where continuity breaks. These are called jump discontinuities.
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Q1: What are the three conditions required for a function to be continuous at a point?
A function is continuous at point a if three conditions are met: the function is defined at a, the limit exists as x approaches a, and this limit equals the function's value at a. These conditions ensure the graph has no breaks, holes, or jumps at that point.
Q2: What is a removable discontinuity and how can it be fixed?
A removable discontinuity occurs when the two-sided limit exists but the function is undefined or misdefined at that point. You can resolve it by redefining the function value to match the limit. For example, as x approaches 1, outputs approach 2, so redefining the function at x = 1 to equal 2 removes the discontinuity.
Q3: How do jump discontinuities differ from infinite discontinuities?
Jump discontinuities occur when left-hand and right-hand limits are both finite but unequal, causing sudden changes in the graph. Infinite discontinuities happen when the function approaches infinity or negative infinity near a point, such as f(x) = 1/x² at x = 0, where the function grows without bound.
Q4: What does a vertical asymptote indicate about function continuity?
A vertical asymptote occurs where a curve shoots upward or downward infinitely without touching a vertical line. This indicates an infinite discontinuity at that point, meaning the function is not continuous there because it grows without bound as x approaches that value.
Q5: How can you determine if a function is continuous by examining its graph?
A function is continuous at a point if two points approaching from opposite directions meet smoothly without breaks or jumps. Smooth movement along the curve indicates continuity. If the graph has holes, vertical jumps, or vertical asymptotes, the function is discontinuous at those locations.
Q6: What is one-sided continuity and when does it apply?
One-sided continuity occurs at interval endpoints where you only check continuity from one direction. For continuity on a closed interval, the function must be continuous at all interior points and satisfy one-sided continuity at the interval's endpoints, ensuring no breaks across the entire interval.
Q7: Why is understanding limits essential for determining function continuity?
Limits are fundamental because continuity requires that the limit as x approaches a point must exist and equal the function's value at that point. By evaluating limits, you can identify whether discontinuities are removable or permanent, making limits the foundation for analyzing continuity.