Which statement correctly describes continuity at a point?

Which statement correctly describes continuity at a point?

• A. Continuity at c requires f(c) defined, limit exists, and limit equals f(c)
• B. Continuity at c requires derivative exists at c
• C. Continuity at c requires f be monotonic
• D. Continuity at c requires f be bounded near c

Answer: (A)

Continuity at a point means the function’s value at that point agrees with the limiting value of the function as you approach that point from either side. In precise terms, f(c) must be defined, the limit as x approaches c must exist, and that limit must equal f(c). That exact combination rules out holes or jumps at c and ensures the function behaves nicely there. This is the right description because it captures both having a defined value at the point and the idea that approaching the point from nearby inputs gives you the same value you get by plugging in the point itself. The other statements miss important pieces: requiring a derivative to exist is a stronger condition that implies continuity but is not needed for continuity; a continuous function can fail to be differentiable at a point. Requiring monotonicity is also not necessary for continuity. Lastly, being bounded near the point can hold for functions that are not continuous there (a function can be bounded near c yet have a jump or hole at c), so boundedness near c is not the defining criterion for continuity.

Continuity at a point means the function’s value at that point agrees with the limiting value of the function as you approach that point from either side. In precise terms, f(c) must be defined, the limit as x approaches c must exist, and that limit must equal f(c). That exact combination rules out holes or jumps at c and ensures the function behaves nicely there.

This is the right description because it captures both having a defined value at the point and the idea that approaching the point from nearby inputs gives you the same value you get by plugging in the point itself.

The other statements miss important pieces: requiring a derivative to exist is a stronger condition that implies continuity but is not needed for continuity; a continuous function can fail to be differentiable at a point. Requiring monotonicity is also not necessary for continuity. Lastly, being bounded near the point can hold for functions that are not continuous there (a function can be bounded near c yet have a jump or hole at c), so boundedness near c is not the defining criterion for continuity.