Let f(x) and g(x) be functions, both continuous at x = a. This can help you solve the problem easily and quickly. If you are given a function and you are required to comment on the continuity of the function, then instead of attempting to find the limits you should just figure out the points of the domain where the function is not defined. Lim x→a- f(x) and lim x→a+ f(x) exist and are equal but not equal to f(a).Īt least one of the limits does not exist. Lim x→a- f(x) and lim x→a+ f(x) exist but are not equal. Hence, a function is said to be discontinuous in any of the following situations: Inverse of a discontinuous function can be continuous. Rational functions at every point except where its denominator is zero. The concept of continuity is always with respect to a particular domain.Ĭertain functions which are always continuous in their domains include: left hand limit is equal to the value of the function at that point and is equal to the right hand limit of the function at that point.Ĭontinuity at the point x = a ⇒ existence of limit at x = a but not the converse.Ĭontinuity at the point x = a ⇒ f is well-defined at x = a but not the converse.ĭiscontinuity at x = a is meaningful provided the function has a graph in the immediate neighborhood of x = a not necessarily at x = a. So the condition for continuity if function at x = a can be defined as L.H.L.= f(a) = R.H.L. The first graph is of a continuous function while the second and third graphs denote discontinuous function. The above three figures clearly illustrate the concept of continuity of a function. If f(x) is not continuous at x = a, we say that f(x) is discontinuous at x = a.įor the function to be continuous at any point x = a, the function must be defined at that point and limiting values of f(x) when x approaches a, is equal to f(a).Ĭontinuity means the function should not have any break or sudden jump at any point in the given domain. = f(a) = value of the function at a i.e. lim x→a f(x) = f(a) The figure drawn on the left shows the graph of a continuous function since it can be made in one flow without lifting the pen and without any sort of breaks.Ī function f(x) is said to be continuous at x = a if lim x→a - f(x) = lim x→a + f(x) = f(a) In simple words, the graph of a function is said to be continuous at x = c if while travelling along the graph of the function and in fact even crossing the point x = c whether from Left to Right or from Right to Left, one does not have to lift his pen. Both the concepts are quite interrelated and limits lays the groundwork for the concept of continuity. Concepts of Physics by HC Verma for JEEĬontinuity of some of the common functionsĬontinuity of functions in which signum function is involvedĬontinuity is also an important component of differential calculus. IIT JEE Coaching For Foundation Classes.
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