There are different answers to this question depending on which conflict you are talking about.

During the First World War, the front line was indeed a line, a continuous line that stretched from the Swiss border to the North Sea in a largely unbroken line of trenches, breastworks and forts some 710 kilometers long from 1914 to 1918. Due to the nature of trench warfare and the inability to penetrate the trench line until the last months of the war the full length was manned by men. When an attack did take place and any advance was made, with a few exceptions, the trench lines bent and flexed to accommodate the taking of ground but did not break until 1918. When you see animations of WW1 front lines moving about, the depiction of solid lines is not far from reality.

The Second World War was different. Advances in motorised technology such as reliable tanks, trucks, personnel carriers and motorised artillery plus communication technology and aircraft meant that static, fixed fortifications were rarely permanent. Depictions of Second World War battles with fronts represented as solid lines are only intended to represent the approximate front lines held by each side but do not mean that the entire line was held at all times. At most, small untis of sentries and recon units would occupy areas along the axis of a main advance and these lines would be permeable if a soldier from one side decided to take a stroll into a region occupied by the enemy. While he might go unnoticed, he couldn't really cause a heck of a lot of damage before he was seen, captured or killed. This could not be accomplished by large formations of infantry and vehicles. First, their advance would likely be spotted from the air if they were German. Second, large formations of men and vehicles are quite difficult to hide in countryside that is filled with men looking for them. Third, the ground has to be able to support an attack by large formations of men and vehicles. Tanks might be capable of off road travel but the trucks carrying their fuel and ammunition need roads. While a surprise attack by German panzers might succeed in penetrating what is considered to be the front line of the enemy, without resupply, they are going to quickly become susceptible to counter attack. This restricts the avenues of attack and advance. Additionally, the more mobile nature of warfare during the Second World War meant it wasn't necessary to man an entire length of frontline. If an enemy did attack, a counter attack by a unit in the vicinity but not necessarily exactly where the attack took place could be launched. Striking into an undefended part of an enemies line also leaves open the possibility of counter attack on the attacking forces flanks. Perhaps the best example of trouble faced by attacking forces (particularly German) during the Second World War is the Ardennes Counteroffensive, better known as the Battle of the Bulge. Attacking a relatively thinly held sector of the front, the German forces managed to make successful advances throughout the Ardennes region of France and Belgium. Poor weather and supply issues dogged the Allies, restricting air cover and reconnaissance as well as stalling their advance. However, despite initial gains, Allied airpower was eventually able to play a role and the inadequacy of German supply lines forced the offensive to a close, forcing the Germans to resume defensive operations for the remainder of the war.