Defining Continuity at a Point

A function f is continuous at x = c if f(c) exists, lim_{x→c} f(x) exists, and lim_{x→c} f(x) = f(c).

Limits10–12% of exam

Understand It

Ace It

Context

What this topic is and why it exists

Continuity at a point is defined by a precise condition: for a function f to be continuous at x = c, three things must happen.

First, the function value f(c) must exist.

Second, the limit of f(x) as x approaches c must exist.

Third, these two values must be equal:

lim_{x→c} f(x) = f(c)

This mechanism ensures the function behaves predictably at that point, without jumps or holes.

The reason it works this way is that continuity is about seamless transition; the function doesn't suddenly change value as you reach x = c.

The trap here is thinking a function is continuous just because it's defined everywhere.

A function can be defined at every point and still have discontinuities if these three conditions aren't met.

The mistake often happens when you assume the existence of a limit without checking if it matches the function's value at the point.

Missteps here can ripple through your understanding of derivatives and integrals, which are built on this foundation.

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