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Left-hand continuity looks at the behavior of a function as we approach a specific point from the left. For a function to be continuous from the left at a point, the limit from the left must equal the function's value at that point. In mathematical terms, a function \( f(x) \) is left continuous at \( x = c \) if \( \lim_{x \to c^-} f(x) = f(c) \). However, if the function is not defined for values less than \( c \), left-hand continuity cannot be evaluated. For example, with the function \( f(x) = \sqrt{x - 1} \) which is defined only for \( x \geq 1 \), there are no values for \( x < 1 \). Therefore, discussing left-hand continuity at \( x=1 \) doesn't make sense, because the function doesn't exist to the left of 1.

Continuous functions have no breaks, jumps, or holes at any points in their domain. For a function to be fully continuous at a point \( x = c \), the left-hand limit, right-hand limit, and the function's value at that point must all be equal. In formal terms:

• \( \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) \) which means the left and right limits at \( c \) must exist and be equal.
• \( \lim_{x \to c} f(x) = f(c) \), ensuring the limit matches the function's value at \( c \).

For instance, in our example, \( f(x) = \sqrt{x-1} \) is continuous from the right at \( x=1 \), as the right-hand limit and function value are both zero, but it cannot be left continuous at this point due to the absence of \( x < 1 \) values.

Graphical representation helps visualize a function’s behavior and its continuity. By graphing, we can easily observe if a function forms an unbroken or smooth curve. For the function \( f(x) = \sqrt{x - 1} \):- It begins at point \((1, 0)\). - Since \( f(x) \) is only defined for \( x \geq 1 \), the graph starts at \( x=1 \) and rises slowly, curving gently away from the origin. This behavior is similar to the right half of a sideways-opening parabola.Visualizing this graph, we see that there is no discontinuity for \( x \geq 1 \). The continuity from the right is apparent as there is a smooth, continuous curve. However, the graph stops and does not exist for \( x < 1 \), illustrating the absence of left continuity for the function.