Continuity at a Point

Continuity at a Point

To check whether a function is continuous at a specific point, compare the function’s value there with the value approached from nearby points.

1) Find the three required quantities

For a point (a), determine:

• (f(a))
• (\lim_{x\to a^-} f(x))
• (\lim_{x\to a^+} f(x))

If both one-sided limits exist and are equal, the two-sided limit exists.

2) Apply the continuity test

The function is continuous at (a) when all three match: [ \lim_{x\to a} f(x)=f(a). ] If the limit does not equal the function value, or if one-sided limits disagree, the function is not continuous at that point.

3) Simplify carefully

When the expression contains fractions, roots, or piecewise parts, simplify first if possible, then substitute. This helps avoid mistakes such as dividing by zero or choosing the wrong branch.

4) Check your result

A quick verification is to compare the limit with the actual value at the point. If they are equal, the point is continuous; otherwise, it is not.

In problems with exact answers, keep expressions in simplest exact form rather than using decimal approximations.