Continuity & Types of Discontinuity

Why we need to learn this

• Continuity tells us if a function’s graph is smooth without breaks, jumps, or gaps.
• It is the backbone of calculus: derivatives and integrals only work properly on continuous functions.
• In real life, continuity models things like motion, temperature, and signals — because nature doesn’t “teleport” suddenly.

How it came

• The idea of smoothness in functions started with Newton and Leibniz (1600s) when they developed calculus.
• It was made precise in the 1800s by Cauchy and Weierstrass, who gave the first rigorous definitions using limits.

Who came up with this idea

• Newton & Leibniz → practical idea of continuous curves in calculus.
• Cauchy → first gave formal definition of continuity using limits.
• Weierstrass → made it even more rigorous with the famous “epsilon–delta” definition.

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Types of Discontinuity (When continuity fails)

1. Point Discontinuity (Removable)

• A single hole in the graph.
• Function is not defined at one point, but the limit exists.
• Example:

• f(x) = (x^2-1)/(x-1), undefined at x=1
• Fixable by redefining the function at that point.

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2. Jump Discontinuity

• The graph “jumps” from one value to another.
• Left-hand limit ≠ Right-hand limit.
• Common in piecewise functions (like step functions).
• Example: cost function where bus fare suddenly jumps from ₹10 to ₹15 after 5 km.

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3. Infinite Discontinuity

• Function heads toward infinity near a point.
• Vertical asymptotes appear.
• Example:

f(x) = 1/x, discontinuous at x=0

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4. Endpoint Discontinuity

• Happens at the edges of the domain of a function.
• If limit does not exist at an endpoint, function is not continuous there.
• Example:

f(x) = sqrt{x}, domain [0, infinity)

At x=0, we check only right-hand limit (since negative side doesn’t exist).

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✅ In short:

• Point → a missing hole.
• Jump → sudden shift.
• Infinite → goes to ±∞.
• Endpoint → break at the boundary of the domain.