If lim_{x→c} f(x) exists and equals L, and f(c) = L, what does this imply about f at c?

If lim_{x→c} f(x) exists and equals L, and f(c) = L, what does this imply about f at c?

• A. continuous at c
• B. differentiable at c
• C. bounded near c
• D. increasing near c

Answer: (C)

Continuity at a point. If the limit of f(x) as x approaches c exists and equals L, and the value f(c) is also L, then f is continuous at c. The function approaches the same finite value from nearby points and actually takes that value at c, so there’s no jump or gap. This is exactly what continuity means at the point c. Note that differentiability isn’t guaranteed by continuity alone, and whether the function is increasing near c isn’t determined by this information.

Continuity at a point. If the limit of f(x) as x approaches c exists and equals L, and the value f(c) is also L, then f is continuous at c. The function approaches the same finite value from nearby points and actually takes that value at c, so there’s no jump or gap. This is exactly what continuity means at the point c. Note that differentiability isn’t guaranteed by continuity alone, and whether the function is increasing near c isn’t determined by this information.